Fig. 9

Possible ranges of the effective chamber compressibility (\(\kappa _\text{ch}\)) for \(\Delta V_\text{E}/\Delta V_\text{G} = 2.69\) and 2.33 (values of the 2011 and 2018 eruptions, respectively), \(p = 150\)–300 MPa, and \(\rho _\text{e} = 1500\), 2000, and 2500 kg m\(^{-3}\) under the assumption that \(n>0\) or \(n=0\). In the analysis of \(n > 0\) for \(n_\mathrm {CO_{2}}^0 = 0.25\)\(-\)1.4 wt%, we assumed three conditions for \(n_\mathrm {H_{2}O}^0\) (2, 4, and 6.2 wt%). The results for \(\Delta V_\text{E}/\Delta V_\text{G} = 2.69\) (2011) and 2.33 (2018) are shown by solid and dashed bars, respectively. For the results with \(n = 0\), the minimum and maximum ends of the bar correspond to \(\kappa _\text{ch}\) values of \(\Delta V_\text{E}/\Delta V_\text{G} = 2.69\) (2011) and 2.33 (2018), respectively. On the right side of the figure, we show the range of \(\kappa _\text{ch}\) estimated from the rigidity of the surrounding host rocks (\(\mu\)) calculated by using P-wave velocity (\(v_\text{p}\)) beneath Shinmoe-dake (Tsutsui et al. 1996) under types of the chamber shape (\(r_\text{v}/r_\text{h} = 1\) for sphere and \(r_\text{v}/r_\text{h} = 0.1\) for oblate spheroid; see Fig. 10). The value of d/r, defined in Fig. 10, varies from 1 to 10