An inflating horizontal penny-shaped sill (Fialko et al., 2001) with a radius of 375 m at a depth of 50 m from the surface beneath Jigokudani valley was found from vertical displacements during the period from September 2020 through September 2021 detected from leveling surveys (Hotta et al., 2023). This inflating sill also applicable to the vertical displacements during the period from September 2021 through September 2022, but it cannot explain horizontal displacements detected by GPS observations (Hotta, 2022).
Therefore, instead of the penny-shaped sill model, a dislocation source (Okada, 1992) was applied in the present study. Assuming Poisson’s ratio of 0.45 (Hotta et al., 2022), the optimal parameters were searched using Adam (Kingma and Ba, 2015) as follows. For a parameter x, an initial value x0 was set randomly. The parameter was then updated \(t\) times sequentially as
$${x}_{t}={x}_{t-1}-\eta \frac{{\widehat{v}}_{t}}{\sqrt{{\widehat{s}}_{t}+\epsilon }} (0<\eta <1)$$
1
,
where
,
$${v}_{t}={\beta }_{1}{v}_{t-1}+\left(1-{\beta }_{1}\right)\frac{\partial }{\partial x}f\left({x}_{t-1}\right)$$
3
,
$${s}_{t}={\beta }_{2}{s}_{t-1}+\left(1-{\beta }_{2}\right)\frac{{\partial }^{2}}{\partial {x}^{2}}f\left({x}_{t-1}\right)$$
4
,
$${\widehat{v}}_{t}=\frac{{v}_{t}}{1-{\beta }_{1}^{2}}$$
5
,
$${\widehat{s}}_{t}=\frac{{s}_{t}}{1-{\beta }_{2}^{2}}$$
6
,
and where \(f\) is an evaluation function to be minimized. In the present study, the evaluation function \(f\) was set to be a weighed residual sum of square, i.e.,
$$f=\sum _{i}{\left(\frac{{u}_{z,i}^{obs}-{u}_{z,i}^{cal}}{{\sigma }_{z,i}}\right)}^{2}+\sum _{j}\left[{\left(\frac{{u}_{x,j}^{obs}-{u}_{x,j}^{cal}}{{\sigma }_{x,j}}\right)}^{2}+{\left(\frac{{u}_{y,j}^{obs}-{u}_{y,j}^{cal}}{{\sigma }_{y,j}}\right)}^{2}+{\left(\frac{{u}_{z,j}^{obs}-{u}_{z,j}^{cal}}{{\sigma }_{z,i}}\right)}^{2}\right]$$
7
,
where \({u}_{z,i}\) is vertical displacement at i-th leveling benchmark; \({u}_{x,j}\), \({u}_{y,j}\), and \({u}_{z,j}\) are east-west, north-south, and vertical displacements at j-th GPS station, respectively; the subscripts obs and cal represent observed and calculated values, respectively; and σ represents standard deviation of the observation. Constants \({\beta }_{1}\), \({\beta }_{2}\), and \(\epsilon\) are set to be 0.9, 0.999, and 1 × 10− 6, respectively. Ranges of initial and \(\eta\) (a learning rate) values for each parameter were set as shown in Table 1. The updates of parameters were done 1 × 106 times. To avoid a local minimum, the analysis was repeated several times.
Table 1
Range of initial and η values of analysis using Adam (Kingma and Ba, 2015; see text) for the model parameters and their optimal values with uncertainties (square brackets).
| Parameter | Range of initial value | η value | Optimal value |
| N-S location [m] | -500 through 0 | 0.6 | -235 [-239, -232] |
| E-W location [m] | -500 through 0 | 0.6 | -376 [-379, -374] |
| Depth from the surface [m] | 0 through 100 | 0.1 | 6.8 [ 5.7, 8.5] |
| Length [m] | 1 through 1000 | 0.3 | 348 [296, ∞] |
| Width [m] | 1 through 1000 | 0.3 | 1.4 [1.1, 1.7] |
| Dip [degree] | 0 through 90 | 0.6 | 86.7 [ 80.2, 92.6] |
| Strike [degree] | 0 through 360 | 0.6 | 239 [238,241] |
| Opening [cm] | 0 through 50 | 0.05 | 36.2 [27.0, 45.4] |
| The origin of the horizontal coordinate is BM. 1 (137.597735° E, 36.58748187° N). The uncertainties are the 99% confidence intervals estimated from an F-test (Árnadóttir and Segall, 1994), and “∞” means cannot be constrained. The position is at the center of the top edge of the fault. The dip is clockwise from the horizontal, and the strike is clockwise from the north. |
The obtained optimal values are shown in Table 1, and the location of the crack and a comparison of the observed and calculated vertical and horizontal displacements are shown in Fig. 2. A crack with a length of 348 m, a width of 1.4 m and a dip of 86.7° is located at a depth of 6.8 m near Koya jigoku and the new fumarolic area, which has recently become highly activated. The strike of the crack is N239°E. The opening of the crack of 36.2 cm yields a volume increase of 173 m3. Although there are some differences between observation and calculation, overall deformation can be broadly explained by the obtained crack opening.
