Applying an FEM, a shallow spherical source with a radius of 50 m was obtained beneath Shin-dake crater at a depth of 400 m asl, i.e., at a depth of approximately 100 m below the crater bottom. According to Lisowski (2007), strain ΔP/μ (where ΔP is internal pressure change; and μ is modulus of rigidity) ranges from 10−3 to 10−1, and the higher value is beyond the typical elastic limit of crustal rocks. The obtained source is within the layer with a modulus of rigidity of 4.52 GPa. The internal pressure change of + 361 MPa yields corresponding strain ΔP/μ of 0.0799. This value is within the range from 10−3 to 10−1, but possibly beyond the typical elastic limit of crustal rocks. Before the eruption on August 3, 2014, surrounding rocks of the source might have shown plastic behavior rather than elastic one. The eruption might have caused when the strain reached their fracture point.
We tested a Mogi model (Mogi 1958), as the previous studies did, to the deformation during January 2006–April 2014 for the purpose of comparing with the result of an FEM. We corrected the topographical effect by adding the elevations of the benchmarks to the source depth (Williams and Wadge 1998). We searched an optimal set of model parameters by a grid search method. The obtained optimal values and 95% confidence intervals for each parameter and the minimum f value are shown in Table 3, and source location and comparison of observed and calculated relative displacements with respect to KUCG are shown in Fig. 9. An inflation source was obtained beneath Shin-dake crater at a depth of 450 m asl, similarly to that of the result using an FEM. Although the Mogi model can explain crater-centered radial horizontal displacements around Shin-dake crater or small displacement at the foot, there are some differences between observation and calculation. Especially, westward displacement at KUC14 cannot be explained by Mogi model. The minimum f value for FEM model is approximately 12% smaller compared to that of Mogi model. Furthermore, its volume increase of 25,400 m3 yields its radial increase of 18 m if the original radius is zero (point source), which is too large under the assumption of Mogi model at extremely shallow depth.
Figure 9a seems to indicate that a NNW-SSE striking dike with a dip toward the west may also explain the deformation. Therefore, we applied the dislocation model for a half-infinite homogeneous elastic solid assuming a Poisson’s ratio of 0.25 (Okada 1992). We searched an optimal set of model parameters similarly to Mogi model. The obtained optimal values and 95% confidence intervals for each parameter and the minimum f value are shown in Table 3, and source location and comparison of observed and calculated relative displacements with respect to KUCG are shown in Fig. 10. The obtained dike can explain crater-centered radial horizontal displacements around Shin-dake crater or small displacement at the foot, and it provided the lowest f value among three models we applied. However, this model still cannot explain westward displacement at KUC14. In addition, this model cannot explain the tilt change at SDN tilt station before the eruption on August 3, 2014 as we will discuss later.
The location of obtained spherical deformation source using an FEM is similar to that obtained by previous studies (e.g., Iguchi et al. 2007) or our result based on Mogi model. Our result using an FEM, however, obtained the shape of the source as a sphere with a radius of 50 m (i.e., a diameter of 100 m), and the source can explain overall deformation, differently to the previous results or our result based on Mogi model. Especially, the location of KUC14, which showed a westward displacement, is a steep topography near the cliff, and should be highly affected by the topography preventing its reproduction by Mogi model.
Hypocenters of shallow volcano-tectonic (VT) earthquakes in 2006 estimated by Triastuty et al. (2009) are distributed at depths of 200–600 m asl beneath Shin-dake crater and coincide with the size and location of the obtained deformation source. They suggested that shallow VT earthquakes were caused by the intrusion of hydrothermal fluids into cracks beneath the crater and extensional stress resulted by the shallow ground inflation source. Accumulation of strain in the rocks around the pressurized deformation source may be one of the causes of the shallow VT earthquakes.
Kanda et al. (2010) obtained a two-dimensional resistivity model by audio-frequency magnetotellurics (AMT) method. They found two conductive regions: one is a shallow and thin conductive layer near the surface beneath the summit area; the other is a deep and thick layer at depths of 200–800 m bsl. They inferred that these regions were conductive clay minerals which are formed by the elevated temperature beneath the area around the current active craters or corresponding to past edifices of Furu-dake and No-ike. Since such clay-rich layers are considered to be impermeable, meteoric water may have been accumulated around the upper boundary of deeper conductive layer and may have formed an aquifer. The obtained deformation source is located between the two clay-rich layers. High-temperature volcanic gas or hydrothermal fluid has been continually supplied from deeper part toward the aquifer, and pressurized rock around the aquifer may have produced demagnetization as Kanda et al. (2010) noted. The obtained deformation source may be corresponding to the pressurized part of the aquifer.
We also applied the obtained spherical deformation source to the precursor tilt change at SDN tilt station located at north rim of Shin-dake crater, before the eruption on August 3, 2014 (Fig. 11a). The tilt change started at around 11:00 AM (local time; the same hereinafter), approximately one and half hour before the eruption. At around 12:00 PM, tilt change accelerated, and the eruption occurred at 12:24 PM. In total, uplift of southwest direction with tilt change of approximately 12.2 μrad was observed. This tilt change can be explained by pressure increase of 6 MPa in the obtained spherical deformation source (Fig. 11b). This internal pressure increase yields volume increase of 520 m3. In 1.5 h before the eruption, 520 m3 of volcanic materials were considered to be supplied toward shallower. We also tried to apply the obtained dike, but this model cannot explain the tilt change at its optimal opening (5 cm) as shown in Fig. 11b.
Yamamoto et al. (2017) obtained the depth of inflation source during the period between August 2014 and March 2015 to be 7.0 km bsl beneath Shin-dake crater from the data of precise leveling surveys. They obtained volume increase as + 2.6 × 106 m3, two orders of magnitude greater than our result during the period between January 2006 and April 2014. A cross-sectional profile of the Kuchinoerabu-jima with deformation sources during the periods between January 2006 and April 2014, and between August 2014 and March 2015, with VT earthquakes zone is described in Fig. 12. Before the eruption on August 3, 2014, volcanic gas or hydrothermal fluid accumulated from deeper toward shallower at a depth of 400 m asl which caused VT earthquakes beneath Shin-dake crater. On the other hand, magma may have accumulated at a deeper part before the eruption on May 29, 2015, and its amount was two orders of magnitude greater than that accumulated toward shallower part before the eruption on August 3, 2014. Besides, magma migration from this deeper reservoir toward shallower may have occurred during the period between December 2014 and January 2015 as Iguchi et al. (2017) indicated from baseline shortening between eastern roof of Shin-dake and Yakushima which cannot be explained by the inflation of deeper source (Yamamoto et al. 2017) as well as increase in SO2 emission during this period.