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Bayesian inference and interpretation of centroid moment tensors of the 2016 Kumamoto earthquake sequence, Kyushu, Japan
 Miroslav Hallo^{1}Email authorView ORCID ID profile,
 Kimiyuki Asano^{2} and
 František Gallovič^{1}
 Received: 4 July 2017
 Accepted: 15 September 2017
 Published: 25 September 2017
Abstract
Keywords
 Tectonics
 Inverse theory
 Waveform source inversion
 Centroid moment tensors
 Strong motion data
 2016 Kumamoto sequence
 Foreshocks and aftershocks
 Futagawa and Hinagu faults
Introduction
This earthquake sequence occurred along the Futagawa–Hinagu fault system, which is one of the major active fault zones in Kyushu. This shear zone is considered to be western extension of the Median Tectonic Line (MTL), the largest tectonic line in southwestern Japan (e.g., Okada 1980; Kamata and Kodama 1994). The MTL is an active rightlateral strikeslip fault which originates at Honshu Island and transects whole Shikoku Island. Extension of the MTL transects Kyushu at NE–SW direction, with evidence of rightlateral strikeslip and extensional movements (Kamata and Kodama 1994). An area in the central part of Kyushu, located north of the shear zone, is called the Beppu–Shimabara graben. It is characterized by many normal faults formed in the N–S extensional stress regime. According to Kamata and Kodama (1999), this extension can be related to the effect of the Ryukyu Trench and convergence of the Philippine Sea slab, where it possibly induces seafloor spreading at the Okinawa Trough. The tectonic stress in Kyushu has large spatial heterogeneities (Matsumoto et al. 2015). At Kumamoto area, the minimum principal stress σ _{3} (extension) is in the N–S direction, and the maximum principal stress σ _{1} has similar size as σ _{2} (Matsumoto et al. 2015); therefore, strikeslip and also normal faults are expected under such stress regime.
The mainshock created more than 30kmlong system of coseismic surface ruptures along the Futagawa–Hinagu fault system (e.g., Kumahara et al. 2016; Toda et al. 2016; Shirahama et al. 2016) terminating in the Aso volcano caldera (e.g., Lin et al. 2016). The surface coseismic ruptures were dominated by rightlateral strikeslips with a secondary normal faulting component. The normal faulting was dominant especially in the northeast part of the rupture zone (e.g., Toda et al. 2016). Finite source models for the mainshock were inverted from strong motion records (e.g., Asano and Iwata 2016; Kubo et al. 2016; Hao et al. 2017; Kobayashi et al. 2017; Yoshida et al. 2017). The inferred models suggest that the M _{JMA}7.3 event started near the intersection of the Futagawa and Hinagu faults by rightlateral strikeslip movement; then, the rupture propagated to the NE along the Futagawa fault as strikeslip with a normal faulting component. Finite source models and also static slip models from geodetic data (Himematsu and Furuya 2016; Fukahata and Hashimoto 2016) are consistent with field measurements of coseismic surface ruptures. The source models introduce two or three segments of the mainshock rupture, but their physical relation to the foreshock ruptures remains unclear because of the complex tectonic settings of the intersection of the Futagawa and Hinagu faults.
This study focuses on the point source models of the significant foreshocks and aftershocks of the Kumamoto sequence. The centroid moment tensors (CMTs) for these events were inferred routinely by JMA and by National Research Institute for Earth Science and Disaster Resilience (NIED). However, these solutions are not supplemented by their uncertainty, which is required when trying to draw conclusions on the physical relations of ruptures in such case of complex tectonic settings. Here we infer CMTs of the significant foreshocks and aftershocks and their uncertainty by our novel fullwaveform inversion from strong motion data. The results are then interpreted in a seismotectonic framework.
Methods
Problem formulation
Bayesian inference of CMT
Parameter c is a constant normalizing the total tendimensional PDF to unity. Note that the term including prior information on the model parameters (Tarantola 2005, eq. 1.104) is not present in Eq. (3) as it is equal to 1.
In Eq. (6), \( \tilde{\varvec{m}}_{i} \) corresponds to the least squares solution of the six model parameters with misfit \( {\mathcal{L}}_{i} \) from Eq. (5) (Tarantola 2005, eq. 3.40). The associated uncertainties (of the least squares solution \( \tilde{\varvec{m}}_{i} \)) are described by the posterior model covariance matrix \( {\tilde{\mathbf{C}}}_{i}^{{\mathbf{M}}} \) given in Eq. (7) (Tarantola 2005, Eq. 3.41).
Vackář et al. (2017) propose to inspect posterior PDF of any CMT parameters by generating set of random possible solutions from the total tendimensional posterior PDF. This can be achieved by drawing a _{ i } random samples at each space–time grid point from the multivariate normal distribution specified by \( \tilde{\varvec{m}}_{i} \) and \( {\tilde{\mathbf{C}}}_{i}^{{\mathbf{M}}} \). From such ensemble, it is then possible to statistically assess uncertainty of any CMT parameter, including those that are not directly inverted for, but can be derived from the MT components (e.g., strike angle, dip angle, rake angle and DC component percentage).
Accounting for the uncertainty of the velocity structure
Application to the 2016 Kumamoto sequence
List of processed earthquakes from the Kumamoto sequence
Event  JMA hypocentre location  Filter corner frequency  Number of used stations  

No.  M _{JMA}  Date  Time (JST)  Lat.  Lon.  Depth (km)  High pass (Hz)  Low pass (Hz)  
1  6.5  2016/4/14  21:26:35  32.74  130.81  11  0.03  0.07  12 
2  6.4  2016/4/15  00:03:47  32.70  130.78  7  0.03  0.08  12 
3  5.9  2016/4/16  01:46:56  32.86  130.90  11  0.04  0.07  11 
4  5.8  2016/4/14  22:07:35  32.77  130.85  8  0.08  0.14  12 
5  5.4  2016/4/16  01:44:06  32.75  130.76  15  0.15  0.20  10 
6  5.4  2016/4/16  09:48:32  32.85  130.84  16  0.08  0.15  13 
7  5.4  2016/4/16  16:02:01  32.70  130.72  12  0.08  0.15  11 
8  5.1  2016/4/14  23:43:41  32.77  130.83  14  0.08  0.14  10 
9  5.0  2016/4/14  22:38:43  32.68  130.74  11  0.09  0.15  10 
10  4.9  2016/4/16  02:04:11  32.74  130.74  12  0.11  0.15  8 
11  4.8  2016/4/16  07:23:55  32.79  130.77  12  0.12  0.16  9 
Data selection and processing
We use threecomponent waveforms recorded by the KNET, KiKnet and Fnet networks, operated by National Research Institute for Earth Science and Disaster Resilience (NIED), in the distances of 10–60 km from the epicenter (depending on the particular event). The stations are selected based on azimuthal coverage, distance and sufficient signaltonoise ratio in the lowfrequency range. Stations located too close to the epicenter are excluded to comply with the point source approximation, i.e., to avoid station distances for which GFs along the fault differ significantly. Original acceleration data (KNET and KiKnet) and strong motion velocity data corrected for the instrument response (Fnet) are filtered by a bandpass filter and integrated into displacements (the KNET and KiKnet accelerographs have flat transfer function in our target frequency range). The filter corner frequencies are determined empirically by manual inspection and processing of the waveforms (see Table 1). In general, the highpass filter corner frequency is as low as possible in terms of signaltonoise ratio. The lowpass filter corner frequency is limited by corner frequency of the particular earthquake. Waveform data are downsampled after the filtration and integration to the sampling frequency 0.6–1.3 Hz, based on the lowpass filter corner frequency, in order to reduce computational demands.
Velocity model
Application details
 1.
Inference of full CMT, including isotropic component, is performed by inversion from extended set of stations without considering any model or data uncertainty. At this stage, we consider a rough grid of space–time grid points in the search of the nonlinear parameters, with regular grid steps of 1.4 km in all three coordinate directions within 16 km around the hypocenter reported by JMA. Time grid covers 0–4 s after the hypocenter time with regular time grid steps of about 0.3–0.1 s (depending on the particular event).
 2.
We manually inspect the best fit of the synthetic and measured waveforms to reveal stations with unusable signal. The proposed Bayesian inference is intended to deal with unknown velocity model perturbations which are close to the 1D velocity model, but it cannot correct for missing velocity structures or other systematical errors in the waveform data. Hence, stations with very poor fit are excluded from the next step of the processing. Mostly, it is the case of very distant stations or receivers located in the coastal area of the Ariake Sea. The final number of used stations is shown in Table 1.
 3.
We perform the Bayesian inference of CMT taking into account velocity model uncertainty of 10%. Here we consider a denser grid in the search of the nonlinear parameters with regular grid steps about 0.2–0.5 km in all three coordinate directions within 3–5 km around the CMT location inverted in the first step. Time grid covers 0–4 s after the hypocenter time with regular time grid steps of about 0.1 s. Since isotropic component of MT was negligible in all cases, we conserve it at 0%.
As the ensemble of acceptable solutions, we generate 1000 random possible solutions drawn from the tendimensional posterior PDF. The resulting MTs are decomposed into combination of doublecouple (DC) and compensated linear vector dipole (CLVD) sources. Marginal PDFs of selected CMT parameters (marginal histograms) are fitted with Gaussian function defined by its mean and standard deviation σ. We consider 2σ as the estimate of the uncertainty covering halfwidth of a 95% confidence interval of the normal distribution.
Solutions of the CMT inversion
Parameters of the inferred CMT solutions together with their uncertainty in terms of double standard deviation of Gaussian function (2σ) fitted to the respective marginal PDF
No.  CMT location  DC component  DC component uncertainty  

Lat.  Lon.  Depth (km)  M _{ w }  VR (%)  S/D/R (°)  DC (%)  S/D/R (°)  DC (%)  
1  32.780  130.809  8.1  6.1  44  33/82/−155  64  3/7/9  ± 24 
2  32.696  130.768  3.8  6.0  58  212/77/178  87  2/9/12  ± 20 
3  32.862  130.856  8.4  5.7  59  294/37/−48  67  6/4/8  ± 16 
4  32.780  130.823  8.3  5.4  73  29/69/−149  94  2/3/4  ± 10 
5  32.765  130.760  13.0  5.0  45  6/72/−142  79  5/7/10  ± 14 
6  32.860  130.835  10.0  5.2  60  83/62/−71  89  3/2/3  ± 10 
7  32.692  130.716  7.8  5.2  74  68/63/−95  75  3/2/3  ± 8 
8  32.765  130.803  9.9  4.9  54  16/76/−163  92  2/4/4  ± 14 
9  32.679  130.735  8.1  4.9  71  211/66/175  90  2/3/3  ± 8 
10  32.745  130.752  5.7  4.7  53  215/81/−165  83  2/7/8  ± 22 
11  32.800  130.788  5.7  4.6  61  79/29/−104  98  8/2/8  ± 12 
The variance reduction defined by Eq. (9) is high for most events; nevertheless, solutions for two events (Nos. 1 and 5) have it below 50%. In the case of event No. 5, the raw waveforms seem corrupted at very low frequencies; hence, slightly higher frequency band was used (see Table 1), which most likely causes the fit deterioration. The event No. 1 is the strongest foreshock which was shown to consist of two spatially separated asperities by Asano and Iwata (2016). Our inferred CMT solution is located between the asperities; nevertheless, the lower variance reduction is likely related to the fact that the distance of the nearest station is at the margin of applicability of the point source approximation (20 km CMTtosite distance vs. 12 km length of the fault).
The uncertainties of the CMT locations are up to 1.5 km in both horizontal and vertical directions for all events. Moreover, events with high variance reduction have CMT location uncertainty as small as 0.6 km. Table 2 documents that the percentage of the DC source content has spans from 64 to 98%. In most cases, it is characterized by large uncertainty including also possibility of pure shear (DC 100%), and hence, the presence of CLVD component cannot be proved (but also disproved) for most of the inferred CMT solutions. The exceptions are events No. 1, 3 and 7 whose admissible DC values do not exceed 90% even taking the uncertainty into account (64 + 24, 67 + 16 and 75 + 8%, respectively; see Table 2). Hence, we consider these events as having a significant CLVD component. Moreover, event No. 7 has the highest variance reduction in all the events, and hence, we consider its CLVD component particularly well constrained.
CMTs with significant CLVD component
The inferred CMT solutions of the Kumamoto earthquakes were decomposed into combination of DC and CLVD sources. While the DC component has direct physical interpretation in terms of shear faulting, the CLVD component points to possible complexity of the faulting (e.g., Frohlich 1994). Indeed, the nonDC MT can be decomposed into a combination of two or more DC sources. Unfortunately, such decomposition is mathematically nonunique, which brings difficulties to interpretation, and requires some physical constraint.
NonDC MT decomposition into two DC MTs
In such formulation, the major MT is the best DC approximation of a shear seismic source, under additional assumption of preserved main principal stress axis (axis with \( \lambda_{1} \)) of the major and minor MTs.
Note that the scalar seismic moments obtained by Eq. (15) may also have negative values; such solutions are considered unphysical and are thus excluded. Then, we accept solutions with misfit (Eq. (16)) lower than 1%. The result of this approach is a set of possible decompositions of nonDC MT into two DC MTs with prescribed main axis difference being less than 20°.
Application of the nonDC MT decomposition
Three of the inferred MTs of the Kumamoto earthquakes (events No. 1, 3 and 7) have significant CLVD component. Two of them (events No. 1 and 7) have Taxis as the main principal stress axis; hence, we analyzed those two in detail. The first analyzed earthquake is the strongest M _{JMA}6.5 strikeslip foreshock, and the second event is M _{JMA}5.4 normal faulting aftershock. The decomposition of these nonDC MTs into major and minor DC sources is depicted in Fig. 7a, b. The summed MTs in Fig. 7 are in perfect agreement with our original nonDC MTs in Fig. 6. In the decomposition with preserved Taxis (Fig. 7a, b, and Appendix 2), the major MT is the best DC approximation of the shear seismic source, while minor MT represents complexity of the faulting (secondary faulting mechanism). It has to be emphasized that the major and minor DC sources cannot be understood as two asperities of an activated fault system; the decomposition is merely mathematical description of a complex earthquake source.
Better understanding can be gained from the examples of possible decomposition provided by the grid search approach (Fig. 7c, d), documenting the nonuniqueness of the decomposition. Nevertheless, all the selected examples are, generally, a combination of a strikeslip and a normal dipslip mechanism (as shown in Fig. 7), which is consistent with the tectonic settings of the intersection of the Hinagu and Futagawa fault zones.
Geometry of the activated ruptures
Event  A (km^{2})  L (km)  R (km)  

No.  M _{JMA}  
1  6.5  155.4  12.5  1.3 
2  6.4  110.3  10.5  1.1 
3  5.9  62.5  7.9  0.2 
4  5.8  28.8  5.4  1.6 
5  5.4  10.3  3.2  0.0 
6  5.4  16.1  4.0  1.7 
7  5.4  16.2  4.0  1.6 
8  5.1  9.8  3.1  1.7 
9  5.0  8.2  2.9  0.3 
10  4.9  5.3  2.3  3.9 
11  4.8  4.9  2.2  2.6 
Discussion
Solutions in seismotectonic framework
Coactivation of the rightlateral strikeslips with normal faulting ruptures through the sequence was introduced also by static slip model from geodetic data (Himematsu and Furuya 2016; Kobayashi 2017). Field investigation by Toda et al. (2016) shows that surface displacements along the previously mapped active fault traces of the Hinagu–Futagawa fault zone are dominated by rightlateral strikeslip surface displacement up to 2 m. A normal surface rupture zone of about 10 km in length dipping to northwest, which is parallel to the Futagawa fault outside the Aso caldera, was also reported by Toda et al. (2016), and its maximum coseismic displacement is also up to 2 m. The normal dipslip aftershocks that occurred along the NW edge of the mainshock rupture had no clear relationship with coseismic surface ruptures; however, minor surface ruptures in downtown of Kumamoto City have been mapped by InSAR (Himematsu and Furuya 2016) and field survey (Goto et al. 2017). The field investigations imply the complex surface phenomena and tectonic settings in this region. Further surveys on imaging causative source faults beneath the surface are necessary to investigate the relationship between the surface ruptures and the geometry of earthquake source faults.

Activity started on April 14th by the M _{JMA}6.5 foreshock close to the intersection of the Hinagu and Futagawa fault zones as rightlateral strikeslip shear movement in the NE–SW direction on fault plane(s) dipping slightly to the ESE (events No. 1 and 4).

Rightlateral strikeslip shear movements continued by simultaneous activity in the northern (dipping to the ESE) and southern (dipping to the WNW) segments of the Hinagu fault zone (events No. 8 and 9).

On April 15th, activity migrated to the southern (dipping to the WNW) segment of the Hinagu fault zone. The activity comprised the second largest foreshock M _{JMA}6.4 (event No. 2).

This was followed by the mainshock on April 16th as rightlateral shear slips complemented by normal dipslip in the Futagawa fault segment in the later phase of the rupture propagation.

Aftershocks in the area of interest were mostly normal dipslip events, spreading along the NW edge of the mainshock rupture (events No. 3, 6, 7 and 11).
Complexity of activated ruptures
The local stress field conditions (e.g., Matsumoto et al. 2015) and seismotectonic settings of the intersection of the Hinagu and Futagawa fault zones imply combination of a strikeslip and a normal dipslip shearing mechanism. Indeed, the source models for the mainshock (e.g., Asano and Iwata 2016; Kubo et al. 2016) suggest simultaneous rightlateral strikeslip shear movement complemented by normal dipslip movement of the Futagawa fault segment. The rupture of the mainshock then has to be complex and segmented to two or more fault planes (as suggested by the mainshock source models).
The analyzed foreshocks and aftershocks are mostly rightlateral strikeslip and a normal dipslip shearing event with insignificant CLVD component and hence may be assumed as single fault plane ruptures. The exceptions are the M _{JMA}6.5 foreshock (event No. 1) and the M _{JMA}5.4 aftershock (event No. 7) with significant CLVD component and Taxis as the main principal stress axis. These two events can be interpreted as a result of complex ruptures composed of rightlateral strikeslip and a normal dipslip fault plane with preserved Taxis (Fig. 7). The assumption of preserved Taxis is in accord the inferred principal stress in Fig. 11, where σ _{3} is stable in N–S direction. Such interpretation of nonDC component is supported by the static slip model for the M _{JMA}6.5 foreshock inferred from InSAR data (Kobayashi 2017), being composed of rightlateral strikeslip displacement on the Hinagu fault segment and normal dipslip displacement on the Futagawa fault segment.
Comparison with routine MT catalogues
The MTs for most of the processed events were also inferred routinely by JMA and NIED institutions. Our and JMA approaches infer CMTs by searching the centroid location in both horizontal and vertical directions. Contrarily, NIED fixes the horizontal centroid location at the (revised) JMA epicenter and searches for the centroid depth only.
Our MTs are compared with solutions from JMA and NIED catalogues in Fig. 6. Our solutions generally agree with both agency MTs in terms of nodal planes angles, while nonDC components agree better with JMA (e.g., see the DC component percentage for the M _{JMA}6.4 foreshock, No. 2 in Fig. 6). This can be related to the use of higher number of records from shorter epicenter distances and searching CMT in both horizontal and vertical directions in our and JMA approaches. Moreover, uncertainty estimate by our approach allows the assessment of reliability of the inferred nonDC component. For example, in cases of events No. 2 and 11 (Fig. 6), the JMA and our solutions exhibit opposite signs of the CLVD component, but this difference is within the estimated uncertainty.
Conclusions
We have presented application of the innovative Bayesian fullwaveform CMT inversion method, which takes into account uncertainty of the velocity model. The approach allows us to reliably assess the uncertainty of the source parameters, which proved to be beneficial in terms of interpretation of the results (statistical significance of selected source parameters). Additionally, we have implemented decomposition of MT with significant CLVD component into two shear MTs with preserved Taxis as a physical constraint.
The methodology has been applied to significant earthquakes from the Kumamoto sequence of April 2016 with M _{JMA} magnitude in range of 4.8–6.5. The quality of the inferred solutions is mostly high as we have used dense network of local to regional receivers. The inferred CMT solutions show systematic spatial and temporal variations. Hence, we have estimated geometry of the major activated ruptures and interpreted them in the seismotectonic framework. Foreshocks imply rightlateral NE–SW strikeslip movements in the Hinagu fault zone. Aftershocks are mostly normal dipslip events spreading along the NW edge of the assumed mainshock fault plane. Moreover, the inferred CMTs with significant CLVD component may suggest a complex source process. These events can be interpreted as a result of complex ruptures composed of rightlateral strikeslip and a normal dipslip fault plane. Our model of major activated ruptures inferred from seismic data is consistent with the local tectonic settings, stress field conditions and geodetic data.
Declarations
Authors’ contributions
MH analyzed the data and drafted the manuscript. KA suggested to analyze 2016 Kumamoto sequence and supported the research by knowledge of the local seismotectonic conditions. FG contributed mainly in methodology and with primary draft corrections. All authors read and approved the final manuscript.
Acknowledgements
The authors thank all operators involved in station maintenance at KNET, KiKnet and Fnet networks. We appreciate relative locations of hypocenters (same as in Kato et al. 2016) provided by A. Kato from Earthquake Research Institute, University of Tokyo, Japan. Generic Mapping Tools (Wessel and Smith 1998) was used to draw the figures. This paper is based on achievements of the collaborative research program (28S02) of the Disaster Prevention Research Institute of Kyoto University. We acknowledge financial support by the Grant Agency of the Charles University project GAUK728916 and Charles University Grant SVV 260447/2017. We are grateful to two anonymous reviewers for their comments that improved the original manuscript.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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