Details of the strain change at the beginning of inflation can be inferred from visual inspection of the time series. Figure 3 shows raw and detrended extensometer data, with daily precipitation data from EBN. A clear change in strain is recognized on each component during 15–17 December 2009. Before and during this change, little precipitation is observed; therefore, the observed strain change is unlikely to be caused by rain-related distortion. The strain change is characterized by extension on E2 and contraction on E1, which we expect when ground deformation is generated by a subsurface pressure source in the azimuthal direction of Shinmoe-dake. This observation suggests that rapid pressurization, on a time scale of a few days, occurred beneath Shinmoe-dake at the beginning of the period of gradual inflation of the magma reservoir inferred from GNSS data (Nakao et al. 2013).

To quantify the strain changes of 15–17 December 2009, the linear trend must be removed from the original time series data. These trends were probably caused by heavy rains during the preceding summer and are unlikely to be related to volcanic deformation. Detrended time series are shown in Fig. 3b. From visual inspection, we estimate that the values of the strain changes in directions E1, E2, and E3 are \(- \;1.6 \pm 0.2\), + \(1.7 \pm 0.3\), and \(- \;0.8 \pm 0.2\), respectively, with units of \(10^{ - 7}\) strain. The strain changes are as large as those observed during each sub-Plinian eruption at Shinmoe-dake in January 2011 (Yamazaki et al. 2013). This suggests that the volume change during the December 2009 event was the same order of magnitude as previous eruptions that occurred during each sub-Plinian eruption.

If we assume that the strain change is generated by a subsurface spherical source in a homogeneous elastic isotropic half-space, the azimuth of the strain event relative to the observatory can be estimated using the ratio of strain changes in the three directions. If we further assume the spherical source can be approximated by a point, the equation for the ground deformation caused by this model has an analytical solution (Yamakawa 1955), with the displacement, *d*, given by

$$d\left( r \right) = \left( {1 - \nu } \right)\frac{\Delta V}{\pi }\frac{r}{{\left( {r^{2} + h^{2} } \right)^{3/2} }}$$

(1)

where *r* and *h* are the distance from the epicenter and source depth, respectively (Fig. 4), \(\nu\) is Poisson’s ratio, and *ΔV* is the volume change. By differentiating this equation, the change in strain is given by

$$f\left( {r,D,I} \right) = A\left( {r,I} \right)\left( {1 - 3\cos^{2} D + \tan^{2} I} \right),$$

(2)

where *D* and *I* represent the azimuth and inclination relative to the extensometer, respectively (Fig. 4), and

$$A\left( {r,I} \right) = \left( {1 - \nu } \right)\frac{\Delta V}{{\pi r^{3} }}\frac{1}{{\left( {1 + \tan^{2} I} \right)^{5/2} }}.$$

When the directions of E1 is set to \(D = D_{1} = 0\) and the azimuth is measured in an anticlockwise sense, the relative azimuths of E2 and E3 are given by \(D_{2} = D_{1} + \pi /2\) and \(D_{3} = D_{1} + 3\pi /4\), respectively. Consequently, the ratios E2/E1 and E3/E1 are given by \(f\left( {r,D_{1} + \pi /2,I} \right)/f\left( {r,D_{1} ,I} \right)\) and \(f\left( {r,D_{1} + 3\pi /4,I} \right)/f\left( {r,D_{1} ,I} \right)\), respectively.

The estimated source direction of the rapid strain change in December 2009 is shown in Fig. 5, together with sources of strain changes observed during the sub-Plinian eruptions and subsequent magma effusion in January 2011. Note that the depths of the two sources can only be compared using inclination angles (Fig. 4). If the depth of one source is greater than another, it will also have a larger inclination, unless the two sources are horizontally distant. Our results suggest that the rapid pressurization was deeper than the shallow magma reservoir that fueled the major eruption. Because a point source is a rough approximation of a magma reservoir, the absolute location may not be correct; however, the relative depths of the two events should be correct if the same point source model and dataset are used. This validates our conclusions regarding the relative depths and sizes of the sources.

The volume change that generated this inflation was large. The strain changes observed in December 2009 are as large as those observed during each sub-Plinian eruption in January 2011 (Yamazaki et al. 2013), suggesting that the volume change was the same order of magnitude as that of each sub-Plinian eruption, which is estimated to be \(1.0 \times 10^{6} {\text{m}}^{3}\) (Table 1 from Yamazaki et al. 2013). If the source of the December 2009 event is deeper than that of the sub-Plinian eruptions in 2011, then the December 2009 volume change would have been even greater. Assuming an epicenter distance of 11.6 km, a source depth of 8–10 km, and a Poisson’s ratio of 0.25 gives a volume change of \(0.6 \times 10^{6} {\text{m}}^{3}\) to \(2.7 \times 10^{6} {\text{m}}^{3}\).