Introduction

A series of earthquakes in Kumamoto and Oita began on 14 April 2016 in the areas along the Oita–Kumamoto Tectonic Line in central Kyushu, Japan (Fig. 1). The Oita–Kumamoto Tectonic Line is an area of high horizontal gravity gradient anomaly (e.g. Tsuboi et al. 1956) located at the boundary between Cenozoic volcanic rocks and Palaeozoic and Mesozoic Erathems (e.g. Kamata and Kodama 1993). This tectonic line consists of many active faults (e.g. Research Group for Active Faults 1991).

Fig. 1
figure 1

Location map of study area. The black dashed line is the Oita–Kumamoto Tectonic Line, and the blue line is the Median Tectonic Line (MTL), which is the largest tectonic line with a right-lateral fault component in south-western Japan. The light grey area with blue dashed line indicates the Hohi volcanic zone (Kamata 1989). The area between ochreous lines is the Beppu–Shimabara Graben (Matsumoto 1979). Red lines indicate the calderas; green lines are active faults (e.g. Research Group for Active Faults 1991), and red stars indicate epicentres of which the magnitude was more than 5 in earthquakes occurring between 14 April and 19 April 2016

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The central Kyushu area is well known for its active volcanoes and calderas such as the Aso and Shishimuta calderas (e.g. Yokoyama 1963; Kubotera et al. 1969; Komazawa 1995; Kamata 1989). This area also includes the Beppu–Shimabara Graben (Matsumoto 1979) in addition to brisk seismic activities with focal mechanisms of lateral faulting, normal faulting or both (e.g. Shimizu et al. 1993; Sudo 1993). The strain field in Kyushu during the last 100 years, which was obtained by geodetic survey, indicates N–S extension (Tada 1984). Matsumoto et al. (2015) reported that the stress condition determined by focal mechanisms in the seismogenic layer, shallower than 30 km, has compressive and tensile axes of WSW–ENE and NNW–SSE directions. They also reported that the seismicity around the shear zone at the southern edge of the Beppu–Shimabara Graben is strike-slip faulting and that in the graben is normal faulting.

In general, subsurface structures play important roles in the understanding and discussion of tectonics of active areas. Frequently employed geophysical survey methods in these volcanic areas include geoelectromagnetic and gravity surveys (e.g. Mogi and Nakama 1993; Komazawa 1995; Kusumoto et al. 1996; Handa 2005). In recent years, gravity gradiometry has been introduced (e.g. Jekeli 1988; Dransfield 2010; Chowdhury and Cevallos 2013; Braga et al. 2014).

This type of survey observes a gravity gradient tensor consisting of six components of three-dimensional (3-D) gradients of gravitation by a causative body and has higher sensitivity than gravity surveys. Various analysis techniques using gravity gradient tensors have been developed and have given excellent results in subsurface structure estimations and edge detections (e.g. Zhang et al. 2000; Beiki 2010; Martinez et al. 2013; Cevallos 2014; Li 2015). In the recent years, techniques estimating the dip of a geological structure boundary by using the gradient tensor of the potential fields have been developed and have shown good results (e.g. Beiki 2013; Kusumoto 2015, 2016; Itoh et al. 2016). The dip of the structure plays an important role in numerical simulations that quantitatively examine tectonics (e.g. Finch et al. 2004; Itoh et al. 2014; Kusumoto et al. 2015).

In this paper, as a quick report, we show the dip distribution of the Oita–Kumamoto Tectonic Line estimated by the technique suggested by Beiki (2013) and Kusumoto (2015). With the exception of some geothermal areas, gravity gradiometry survey has not been conducted thus far in central Kyushu. Therefore, we estimated the gravity gradient tensor by using calculations based on Mickus and Hinojosa (2001) to obtain the tensor from the gravity anomaly, which was used for this study.

Gravity anomaly and gravity gradient tensor

Figure 2 shows the residual Bouguer gravity anomaly map of which the first trend surface was removed from the original Bouguer gravity anomaly by using the least squares method in order to eliminate the effect of the subducting plate. A Bouguer density of 2670 kg/m3 was employed here. In this study, the gravity anomaly database by Komazawa (2004) was employed. Because this database provides mesh data with intervals of 1 km × 1 km, we discuss structures larger than several kilometres.

Fig. 2
figure 2

Residual Bouguer gravity anomaly map of the study area. The contour interval is 2.5 mGal. In this map, the regional linear trend of the Bouguer gravity anomalies such as the effect of the subducting plate was estimated and removed by using the least squares method. The Bouguer gravity anomaly used here is based on the gravity anomaly database (1 km × 1 km mesh data) of Komazawa (2004); a Bouguer density of 2670 kg/m3 was employed

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The figure indicates negative gravity anomalies caused by the Shishimuta and Aso calderas and some tectonic sedimentary basins such as Beppu Bay. In addition, high horizontal gravity gradient belts caused by the Median Tectonic Line and the Oita–Kumamoto Tectonic Line are shown. The Beppu–Shimabara Graben shown in Fig. 1 corresponds to the negative gravity anomaly area in the north-east and to a gravity low area less than about 20 mGal in the south-west.

The gravity gradient tensor Γ is defined on the basis of differential coefficients of the gravity potential (e.g. Torge 1989; Hofmann-Wellenhof and Moritz 2005). Defining g x , g y , g z as the first derivative of the gravity potential along the x-, y-, and z-directions, the gravity gradient tensor Γ is shown as

$$\varGamma = \left[ {\begin{array}{*{20}c} {\frac{{\partial g_{x} }}{\partial x}} & \quad {\frac{{\partial g_{x} }}{\partial y}} & \quad {\frac{{\partial g_{x} }}{\partial z}} \\ {\frac{{\partial g_{y} }}{\partial x}} & \quad {\frac{{\partial g_{y} }}{\partial y}} & \quad {\frac{{\partial g_{y} }}{\partial z}} \\ {\frac{{\partial g_{z} }}{\partial x}} & \quad {\frac{{\partial g_{z} }}{\partial y}} & \quad {\frac{{\partial g_{z} }}{\partial z}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {g_{xx} } & \quad {g_{xy} } & \quad {g_{xz} } \\ {g_{yx} } & \quad {g_{yy} } & \quad {g_{yz} } \\ {g_{zx} } & \quad {g_{zy} } & \quad {g_{zz} } \\ \end{array} } \right].$$
(1)

Here, g z is the gravity anomaly, and g zz is the vertical gradient of the gravity anomaly. The gravity gradient tensor is symmetric (e.g. Torge 1989), and the sum of its diagonal components is zero because the gravity potential satisfies the Laplace equation.

Although the gravity gradient tensor is generally measured by gravity gradiometry (e.g. Lee 2001; Barnes and Lumley 2011; Dransfield and Christensen 2013), the tensor can be obtained from the gravity anomaly data by using calculations shown in Mickus and Hinojosa (2001). Mickus and Hinojosa (2001) showed the following procedures: (1) apply a 2-D Fourier transformation to the gravity anomaly; (2) estimate the gravity potential by integration of the gravity anomaly in the Fourier domain; (3) calculate the gravity gradient components by second-order derivatives of the potential in each direction; and (4) apply a 2-D Fourier inverse transformation to finally obtain all components of the tensor in the spatial domain. Integration and differentiation of a function in the Fourier domain are equivalent to division and multiplication by powers of wave number (e.g. Blakely 1996).

This method is excellent and mathematically sound. However, because the second-order derivatives of the potential in the Fourier domain and Fourier inverse transformation emphasise short wavelength signals including noise, we obtained components of the gravity gradient tensor by numerical differentiation of g x , g y , and g z in the space domain. The components of g x and g y were calculated by the method of Mickus and Hinojosa (2001). The horizontal derivatives of g x , g y , and g z were calculated using simple finite-difference methods (e.g. Blakely 1996). The vertical gravity gradient, g zz , has been computed as g zz  = −(g xx  + g yy ), since the gravity potential satisfies the Laplace’s equation.

In the analysis of subsurface structures using the gravity gradient tensor, the horizontal gravity gradient method is the most widely used for finding structure boundaries such as faults or contacts between different materials (e.g. Blakely and Simpson 1986; Shichi et al. 1992; Kudo and Kono 1999; Yamamoto 2003). The horizontal gravity gradient (HG) is given using the components of the tensor shown in Eq. (1) as

$${\text{HG}} = \sqrt {g_{zx}^{2} + g_{zy}^{2} } .$$
(2)

Figure 3a shows a map of the horizontal gravity gradient. Figure 3b shows areas of more than 25 E (Eötvös: 1 E = 0.1 mGal/km) from which structural boundaries such as the outlines of the Median Tectonic Line, Oita–Kumamoto Tectonic Line, Shishimuta and Aso calderas are extracted. It was found that a series of earthquakes occurred on the high gravity gradient belt. At the western terminations of the Median Tectonic Line and of the Futagawa fault system belonging to the Oita–Kumamoto Tectonic Line, steep gravity gradient areas reach 100 E.

Fig. 3
figure 3

Horizontal gravity gradient map. a Horizontal gravity gradient map of which all data are shown; the contour interval is 10 E (Eötvös: 1 E = 0.1 mGal/km). b Horizontal gravity gradient map of which areas of more than 25 E are shown by coloured contours; the contour interval is 5 E

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Dip estimation of fault or density structure boundary

Because the maximum eigenvector (v 1) of the gradient tensor of the potential field indicates the direction of the causative body (Beiki and Pedersen 2010), Beiki (2013) suggested that the dip of the causative body can be estimated from the respective x, y, and z components (v 1x , v 1y , v 1z ) of the maximum eigenvector, v 1, as follows:

$$\alpha = \arctan \left[ {\frac{{v_{1z} }}{{\sqrt {\left( {v_{1x} } \right)^{2} + \left( {v_{1y} } \right)^{2} } }}} \right],$$
(3)

This equation is valid in estimating the dip of the causative body if the body is a 2-D structure like a dyke as shown in Fig. 4a. Structural characteristics of subsurface structure can be evaluated by a dimensionality index. Since the index is <0.5 if the body causing a gravity anomaly is a 2-D structure, it is recommended that this calculation would be conducted under the condition of a dimensionality index of <0.5 (Beiki and Pedersen 2010; Beiki 2013). The dimensionality index (DI) is defined as (Perdersen and Rasmussen 1990)

$${\text{DI}} = \frac{{ - 27I_{2}^{2} }}{{4I_{1}^{3} }}.$$
(4)

Here, I 1 and I 2 are invariants of the gravity gradient tensor. Each invariant is given by three eigenvalues (λ 1, λ 2, λ 3) of the tensor as follows:

$$\begin{array}{*{20}l} {I_{1} = \lambda_{1} \lambda_{2} + \lambda_{2} \lambda_{3} + \lambda_{1} \lambda_{3} } \hfill \\ {I_{2} = \lambda_{1} \lambda_{2} \lambda_{3} } \hfill \\ \end{array} .$$
(5)
Fig. 4
figure 4

Schematic illustration of the eigenvectors for two-dimensional (2-D) structures such as dykes and faults. a Basic model. In this figure, v 1 is the maximum eigenvectors of the gravity gradient tensor and points to the causative body. The angle α between the surface and the maximum eigenvector is the dip of the causative body. b Fault dip model. A basement consists of an assembly of high-density columns, and the angle, α, indicates the fault dip

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Kusumoto (2015) considered that a basement consists of an assembly of high-density columns (Fig. 4b) and applied this idea to analysis of fault dip. To effectively estimate the dip of the fault or density structure boundary, Kusumoto (2015) recommended that the estimation would be conducted at areas of high horizontal gravity gradient. In this study, the dip of the structure boundary was estimated in the area satisfying the conditions of which tectonic lines are extracted precisely and are 2-D structure, namely the areas of HG ≥ 25 E and DI ≤ 0.5.

Results and discussion

Figure 5 shows the estimated dip distribution of large structure boundaries recognised by high horizontal gravity gradient anomalies. The dips of the Median Tectonic Line and Oita–Kumamoto Tectonic Line ranged from of 30° to 80° and were estimated as high-angle tectonic lines or an assembly of high-angle faults. In addition, the dip of the caldera wall of the Aso caldera was estimated to be 50°–70°.

Fig. 5
figure 5

Distribution of estimated dip of faults or structural boundaries. Dip estimations were conducted for the areas satisfying the conditions (DI) <0.5 and horizontal gravity gradient (HG) ≥25 E

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In part of the Median Tectonic Line, the dip is very high and exceeds 70° in some areas. It is well known that the Median Tectonic Line had moved as a right-lateral fault in the Quaternary; thus, the high dips reaching 70° or 80° are expected. The dips of the Median Tectonic Line become low gradually to the north in the Beppu Bay area. Although Kusumoto (2015) reported that this dip estimation technique tends to underestimate the actual dip in deep parts when applied to normal faults, seismic reflection surveys have confirmed dips of normal faults distributed in Beppu Bay to be 17° and 7° by seismic reflection surveys (e.g. Itoh et al. 2014). Therefore, the dip distribution calculated from the gravity gradient tensor agrees with the observed data.

In part of the Oita–Kumamoto Tectonic Line, the general dip was estimated to be about 65°. Because large-scale seismic survey has not been conducted around this tectonic line, its actual dip is unknown. However, the dip estimated from the gravity gradient tensor agrees with the fault dip obtained from crustal movement due to an earthquake in the Futagawa fault system.

The largest earthquake, M = 7.3, occurred on 15 April 2016 in a series of earthquakes at the centre of the Futagawa fault system; its fault parameters are estimated from crustal movement data observed by GNSS (The Headquarters for Earthquake Research Promotion 2016). According to The Headquarters for Earthquake Research Promotion (2016), the earthquake source fault is right lateral with a normal fault component, and the upper depth, length, width, azimuth, dip, slip angle, and slip amount of the fault are estimated to be 0.1, 27.1, 12.3 km, 235°, 60°, −161°, and 3.5 m, respectively. Figure 5 shows that the dips estimated from the gravity gradient tensor are 30°–40° at the northern side and roughly 55°–65° at the southern side. Because the earthquake source fault dips to the north, and this technique tends underestimate the actual dip in deep part for normal faults, the dips estimated from the gravity gradient tensor were found to agree with the fault dip obtained by crustal movement in the central eastern part of the Futagawa fault system.

From these discussions, it appears that the dip estimation technique using eigenvectors and eigenvalues of the gravity gradient tensor effectively provided the dip of the Oita–Kumamoto Tectonic Line, which we conclude to be generally about 65°. In detail, the dip of the Oita–Kumamoto Tectonic Line tends to be higher towards its eastern end and exceeds 70° at the area connecting to the Median Tectonic Line. The dip in the Futagawa fault system area is relatively low at about 60°. This variation or trend is divided into eastern and western areas by the Aso caldera. Although a series of earthquakes occurred along the Oita–Kumamoto Tectonic Line, the cause or origin and formation processes of this tectonic line might be different in each segment.

Although the tectonics in the western part of the Aso caldera are unknown in detail, the eastern part of the Aso caldera is known as the Hohi volcanic zone (e.g. Kamata 1989), which consists of a half-graben with volcanic activities that began in 6 Ma (e.g. Kamata 1989) and pull-apart basins that were formed after the half-graben formation in about 1.5 Ma (e.g. Itoh et al. 1998). These structures and their formation processes have been restored by numerical simulations assuming high-dip (80°) normal faulting and right-lateral faulting (e.g. Kusumoto et al. 1999). It appears that the formation of the half-graben in 6 Ma plays an important role in understanding the spatial variation of dip distribution because the Oita–Kumamoto Tectonic Line is the boundary between Cenozoic volcanic rocks and Palaeozoic and Mesozoic Erathems.

In this study, we used the existing gravity anomaly database (Komazawa 2004) which compiled 1 km × 1 km mesh data of the Bouguer gravity anomaly obtained by Bouguer density of 2670 kg/m3. Although changes of the mesh size and the Bouguer density vary aspects of some maps, it seems that they would not make serious differences in the general situation. However, to obtain a detailed fault shape and to discuss the cause of a series of earthquakes tectonically, in addition to discussions on change of the Bouguer density, it is effective to use a dense gravity database or to conduct gravity gradiometer survey and seismic reflection survey around fault zones in the future studies.

Conclusion

In this study, we estimated the dip distribution of the Oita–Kumamoto Tectonic Line where a series of earthquakes began on 14 April 2016. For dip estimation, the method using the dip of the maximum eigenvector of the gravity gradient tensor was employed. Because gravity gradiometry survey has not been conducted in the study area, the tensor was obtained by calculations from the Bouguer gravity anomaly. The estimation was conducted in an area satisfying the following conditions: (1) a horizontal gravity gradient larger than 25 E and (2) a dimensionality index <0.5.

We obtained that the dip of the Oita–Kumamoto Tectonic Line is generally about 65°. The fault dip around the largest earthquake of M = 7.3 in a series of earthquakes was estimated to be about 60°, which agrees with the dip of the earthquake source fault obtained by GNSS data analysis.

In addition, we found that the dip distribution of the Oita–Kumamoto Tectonic Line tends to be higher towards its eastern end, exceeding 70° at the area connecting with the Median Tectonic Line. On the other hand, the dip in the Futagawa fault system area is relatively low. This spatial variation of dip distribution has been attributed to the formation of a half-graben with volcanic activities that began in 6 Ma. However, more observations and discussions including numerical simulations are needed to understand these results because details of the tectonics in the Futagawa fault system area are unknown.