### Selection of the optimal model for phase 2

To determine the optimal solution, for each model we evaluated the correction of Akaike’s Information Criterion (c-AIC) values (Sugiura 1978), which is a finite correction of Akaike’s Information Criterion (AIC; Akaike 1973). The WRSS values of the three spherical sources and the dual spherical sources with tensile fault models are 13,408 and 9,140.0, respectively, and the number of free parameters is seven and ten, respectively. These yield c-AIC values of 222.19 and 233.53, respectively. Although the residual is larger than the dual spherical sources with a tensile fault model, the three spherical sources model is better, based on the obtained c-AIC values. In another respect, the volume increase of the obtained tensile fault is nearly 10^{7} m^{3} (/1.83 h), which is too large based on the typical magma intrusion rate of Sakurajima, 10^{7} m^{3}/year (Ishihara 1981). The intrusion rate during the inflation period of October 2011–March 2012 was 1.4 × 10^{7} m^{3}/year (Hotta et al. 2016a). From GPS observations, a rate as high as 10^{7} m^{3} in ~ 2 h was not detected around the November 13, 2017 Minami-dake explosion, as shown in Fig. 8, indicating that the intrusion rate at this time was less than on the order of 10^{7} m^{3}/year. Therefore, we concluded that the three spherical sources model is the optimal model for phase 2.

As for a finite spherical source model, Pascal et al. (2014) noted that the effects of interactions between sources are negligible when the sources are separated by four or more radii. The distance between sources *M*_{S} and *M*_{D} is 3.6 km, that between *M*_{D} and *K* is 4.8 km, and that between *K* and *M*_{S} is 5.7 km. Meanwhile, the optimal radii of *M*_{S}, *M*_{D} and K are 0.1 km, 0.5 km and 0.1 km, respectively. From these distances and radii, these three spherical sources are separated enough to ignore the effects of interactions between them.

### Interpretations of the sources

During the 3 min from the onset of the explosion at 22:07, phase 1, a very shallow deflation source was obtained beneath Minami-dake, similar to previous eruptions at the Minami-dake (Ishihara 1990) and Showa craters (Iguchi et al. 2013). This means that the deflation starts from a shallow part of the magma plumbing system of Sakurajima. This shallow source might represent the uppermost part of the magma conduit from a deeper Minami-dake source to the summit crater, which has been interpreted as a high-pressure gas chamber (e.g., Iguchi 1994). After the peak of the Vulcanian eruption, phase 2, the deeper source beneath Minami-dake deflated while the shallower source inflated. The deep deflation source beneath Minami-dake had been identified based on tilt and strain changes during previous eruptions at the Minami-dake and Showa craters. The obtained depth of 3.3 km bsl was comparable with previous studies (2–4 km), and the horizontal location of the source was similar to the fixed location beneath Minami-dake of previous studies (e.g., Ishihara 1990; Iguchi et al. 2013). In addition to the deeper source, inflation of the shallower source was detected. The shallow inflation source beneath Minami-dake found in the present study was not found in the previous studies. During the Minami-dake eruption including this event, the shallower source is considered to deflate for the first phase of each eruption. In this event, the shallower source inflated after the peak of the explosion probably because the amount of accumulated magma from the deeper source exceeded that of the emission from the Minami-dake crater due to continuous Strombolian eruptions (Fig. 9). In addition to the two Minami-dake sources, we found a deflating spherical source at a depth of 3.2 km bsl at the northeastern flank of Kita-dake. The Kita-dake source might correspond to the source *K* found beneath Kita-dake at a depth of 3.3 km bsl from a combination analysis of GPS, tilt and strain data during the period from October 2011 to March 2012 (Hotta et al. 2016a). The Kita-dake source was also found based on vertical displacement during the period from November 2008 to November 2009, obtained from a precise leveling survey (Yamamoto et al. 2013). Not only the deeper Minami-dake source M_{D} but also the Kita-dake source deflated due to the Minami-dake explosion (Fig. 9). The total deflation volume of sources *M*_{D} and *K* was 88,900 m^{3}, which is comparable with the previous Minami-dake eruptions of 10^{3}–10^{5} m^{3} (Ishihara 1990).

A clear reflector at a depth of 5.8 km bsl beneath Kita-dake, identified as an overcritical PP reflection, was found by seismic experiments during the period from December 2009 to December 2014. The reflection became clear in December 2014, and the reflector was interpreted as a sill (Tsutsui et al. 2016). The source K obtained in the present study is shallower than the reflector, but the horizontal location is similar. The 99% confidence interval of the depth of source *K* is 2.6–5.8 km bsl, as shown in Table 1, and the depths of source *K* and the reflector may overlap considering the uncertainty in the depth of the reflector of ± 90 m (Tsutsui, personal communication on December 17, 2020). If source K and the reflector are the same, the difference in the optimal depth may be due to the source shape, such as a penny shaped source (Fialko et al. 2001), or heterogeneity, which we will discuss for volume changes later. These should be considered closely in future studies.

Recent studies on magma plumbing systems (e.g., Cashman et al. 2017) indicate that magma recharge of upper crustal reservoirs, where silicic melts segregate from mafic magma, grow large magma bodies and become a trigger for eruptions. The mafic magma is intermittently resupplied from deeper levels in the system. As for Sakurajima, mafic magma has been frequently injected since the twentieth century (Nakagawa et al. 2011). The intrusion rate from the Aira caldera to the Kita-dake source is up to 500 m^{3}/day during inflation periods (Hotta et al. 2016a) when the SiO_{2} content of volcanic glass decreases, which indicates the injection of mafic magma (Matsumoto et al. 2013). This causes inflation of the Kita-dake source, which acts as a buffer for increases in eruptive activity (Hotta et al. 2016a), becoming a trigger of magma migration to deeper and shallower Minami-dake sources. As a result, magma is finally emitted from the Minami-dake or Showa crater, causing eruptions.

### Effect of topography and heterogeneity

There are many limitations such as topography and heterogeneity in the homogeneous analytical model we used in the present study (e.g., Del Negro et al. 2009; Gottsmann and Odbert 2014; Hickey et al. 2016; Hotta and Iguchi, 2017). Although realistic modeling using FEM is time consuming, we checked for the effect of volume change for phase 2. We applied the FEM using Flex PDE Professional software version 6.50 provided by PDE Solutions Inc. We set an FEM domain size of 100 × 100 × 50 km^{3} (130.13325–131.18675° E, 31.12906–32.03094° N, 0–50 km bsl). The maximum grid size was set to be 30 per 100 km and the grid size in the domain was set automatically by the software. We set the top of the domain to be a free surface, and the sides and bottom of the domain to be fixed boundaries. Topography was introduced in the area within Sakurajima using digital elevation model (DEM) data with 100 m resolution, and the elevation of the outside area was assumed to be sea level. Heterogeneity of the P-wave velocity was introduced referring to Miyamachi et al. (2013), as 1, 2.5, 3.6, 4.8 and 6 km/s for the depth ranges of 0–0.2, 0.2–1, 1–2, 2–3 and 3 + km from the surface, respectively, as shown in Fig. 10. Assuming a Poisson’s ratio of 0.25 and *V*_{P}/*V*_{S} velocity ratio of 1.73 (e.g., Hotta et al. 2016b), the modulus of rigidity *μ* can be written using the density of ground *ρ* as

$$\mu =\frac{\rho {V}_{\text{P}}^{2}}{3}$$

(5)

Currenti et al. (2007) found the following empirical formula between the density of the ground and P-wave velocity:

$$\rho =1.2861+0.5498{V}_{\text{P}}-0.0930{V}_{\text{P}}^{2}+0.007{V}_{\text{P}}^{3}.$$

(6)

From Eqs. (5) and (6), the modulus of rigidity of the P-wave velocity structure can be calculated to be 1.02, 3.99, 6.84, 10.5 and 15.1 GPa for the depth ranges of 0–0.2, 0.2–1, 1–2, 2–3 and 3 + km from the surface, respectively. Using the FEM domain described above, we first calculated the tilt and strain change due to a pressure change of + 1 MPa inside each source (*M*_{S}, *M*_{D} and *K*) while fixing the horizontal location, depth and radius that were obtained using a homogeneous analytical model. We then searched for the optimal internal pressure change by a grid search method based on the fact that tilt/strain changes are proportionate to internal pressure changes in FEM calculations. The search range and step were set to be − 100– + 100 MPa and 0.01 MPa, respectively. The optimal internal pressure changes of M_{S}, M_{D} and K were + 0.71 MPa, − 1.41 MPa and − 82.98 MPa, respectively. These yielded volume changes of + 450 m^{3}, − 36,200 m^{3} and − 16,700 m^{3}, respectively. The yielded WRSS value of 15,997 was slightly larger than that of our homogeneous analytical model, but the data can be well explained (Fig. 11). All of the yielded volume change amounts were smaller than those obtained using a homogeneous analytical model, especially that for the shallowest source M_{S}, the result for which (+ 450 m^{3}) was outside of the estimated 99% confidence interval of the homogeneous analytical model (600–5000 m^{3}). This indicates that modeling by a homogeneous analytical model overestimates of the volume change, particularly for the shallow source. The weight of ashfall in November 2017 is estimated to be 1.75 × 10^{8} kg. Although six eruptions occurred in November 2017, five eruptions excepting that on November 13, 2017, were extremely small-scale (plume heights were not exceeding 2000 m). Therefore, the weight of ashfall by the November 13, 2017 Minami-dake explosion can be assumed to be nearly 1.75 × 10^{8} kg. Assuming a dense rock equivalent (DRE) density of 2500 kg/m^{3}, the volume of ejected magma is nearly 70,000 m^{3}. Since the total deflation volume of sources M_{D} and K from FEM analysis was 52,900 m^{3} and the inflation of the source M_{S} is very small (450 m^{3}), magma was compressed by up to 25% in these deep sources. The pressure change value of the source *K* is very large. In addition, the 95% confidence interval of the radius of source *K* is 0.1–10 km. This result indicates that the radius of the source *K* can be larger than 0.1 km and therefore the pressure change value would be smaller.

In this study, we checked only for volume change. The location, depth and radius of the source should be investigated in a future study.