### Elastostatic modeling of earthquake deformation

We first calculate the spatial distribution of the volumetric strain changes (the sum of the normal components of the tensor). The volumetric strain change around the magma reservoir may relate to an increasing magma overpressure effect (Walter and Amelung 2007) as a source of eruption. Although the spatial distribution of the magma reservoir beneath Mt. Fuji remains unclear, a seismic tomography study (Nakamichi et al. 2007) revealed a high-Vp/Vs anomaly around 15–25 km depth that is interpreted as a zone of basaltic partial melting. Another study (Kinoshita et al. 2015) used receiver function analysis to show a seismic velocity contrast at 20–30 km depth, which can be related to the bottom of the magma chamber. In addition, over 90 % of (magmatic) low-frequency earthquakes in Fig. 2 occurred in a depth range of 10–20 km. Thus, we focus on the strain changes at a depth of 20 km, treating this as a typical value for the depth of a basaltic magma reservoir beneath Mt Fuji.

Figure 4 shows a map view of volumetric strain changes due to each earthquake scenario. Although we do not know the precise geometry of the magma reservoir, it should be around or within the area below the eruptive fissures. The calculated results show that three earthquakes, (a), (b), and (e), cause compression around the magma reservoir and the other earthquakes, (c) and (d), cause dilation. The compressions are one order of magnitude larger than the dilations. Figure 5 further shows the volumetric strain changes in a vertical cross-section along an extended dike plane (the main pathway of Mt. Fuji’s magma), where we extend the dike plane in Chesley et al. (2012) from 13 km length to 30 km.

Next, we calculate the normal stress changes for the extended dike plane from the strain changes, assuming a rigidity of 30 GPa. The normal stress change perpendicular to the extended dike plane works as an opening force for a magma pathway in terms of fracture mechanics (Pollard and Segall 1987); Bonali et al. (2013) emphasized the importance of this effect. Figure 6 illustrates the normal stress changes due to the five earthquakes. In many cases, the extended dike plane is wholly clamped. Only in the case of (c), the 2011 Tohoku earthquake, is the extended dike plane totally unclamped. As reported by Chesley et al. (2012), case (a) shows that the 1707 Hoei earthquake caused unclamping in a deep region of the dike plane, which could have triggered the Hoei eruption.

### Two hypotheses that explain eruption triggering by elastostatic deformation

We estimated two kinds of elastostatic effects: volumetric strain changes around the basaltic magma reservoir and normal stress changes perpendicular to the extended dike plane (the main pathway of Mt. Fuji’s magma). Because we do not consider material contrasts surrounding the magma reservoir, as in Fujita et al. (2013), the calculated values represent the mean fields of peripheral domains. The magnitudes of the elastostatic effects, around 1 MPa at most, are not negligible beneath Mt. Fuji because the differential stress there may be relatively small. For instance, Araragi et al. (2015) proposed that the maximum horizontal stress is only 1.02× greater than lithostatic pressure. They raised an example of a lithostatic pressure of 51.9 MPa at a depth of 2.0 km, which means the difference between the maximum horizontal stress and the lithostatic pressure would be about 1 MPa at that depth.

A previous study of volcano–earthquake interactions (Walter and Amelung 2007) proposed that dilation of a magma reservoir initiates magma ascent, on the basis of volcano eruption data after M ≥9 megathrust earthquakes worldwide. That is, earthquake-induced dilation (depressurization) released dissolved volatile gases (CO_{2} and H_{2}O), leading to volumetric expansion of a magma reservoir. This process decreases the density and viscosity of the magma, though the latter is perhaps not necessarily as suggested by Bottinga and Weill (1972) and Urbain et al. (1982). This, in turn, drives the ascent of gas bubbles with magma, causing further depressurization. In one possible scenario, this leads to an eruption through a positive feedback loop (Walter and Amelung 2007). Since the process of bubble nucleation under sudden depressurization depends on many factors (e.g., Toramaru 2014), the threshold level and conditions for the beginning of this magma ascent process remain unclear.

From this viewpoint and Fig. 5, only two earthquakes, the Tohoku earthquake (c) and the ISTL earthquake (d), have the potential to have initiated volcanic activity. At present, we know that the 2011 Tohoku earthquake has not yet triggered an eruption at Mt. Fuji. The reason for this is probably that the amount of magma dilation was small (< 1 μ strain) relative to known cases of volcanic eruptions after M ≥9 megathrust earthquakes (Walter and Amelung 2007) or that the magma state was not favorable to drive the ascent for eruption (Fujita et al. 2013). Although the 2011 Tohoku earthquake did not trigger an eruption at Mt. Fuji, one point to be noted is that many earthquakes, including a Mw 5.9 earthquake near Mt. Fuji, followed the 2011 Tohoku earthquake (Enescu et al. 2012; Kumazawa and Ogata 2013). We will statistically analyze these induced earthquakes in a later section. The ISTL earthquake, which occurred in 762 or 841, also did not trigger an eruption at Mt. Fuji; similarly, this is thought to be due to the small amount of magma dilation.

In contrast, Nostro et al. (1998) proposed another hypothesis linking volcanic eruptions with earthquakes at Mt. Vesuvius. They suggested that compression of a magma reservoir and unclamping of a dike (magma pathway) by earthquakes led to eruptions. From Figs. 5 and 6, we found no cases corresponding to this model for Mt. Fuji, but the Hoei earthquake (a) was the closest analog. The 1707 Hoei earthquake compressed the basaltic magma reservoir and partly unclamped the main dike, especially in deeper regions, which might have uplifted sufficient basaltic magma to trigger magma mixing and the Plinian eruption 49 days after the Hoei earthquake (Chesley et al. 2012). The case of the 1854 Tokai earthquake (e) is a similar situation, but unclamping in the deep region of the main dike is almost negligible. This might explain why the 1854 Tokai earthquake did not trigger an eruption at Mt. Fuji. In addition, a recharge time of only ~150 years might be too short, particularly for another Plinian eruption. It is possible that the 1703 Kanto earthquake (b) did not lead to an eruption of Mt. Fuji because of total clamping of the extended dike plane, which tends to suppress magma ascent.

Note that compression of the magma reservoir itself would not be enough to trigger an eruption. In three cases of compressional strain changes around the magma reservoir (Figs. 4a, b, e), the amounts of strain changes are ~ − 3 × 10^{− 6}. When we assume simple spheres for the geometry of the magma reservoir, with radii of 1 and 5 km, the volume changes of the magma reservoir are ~ − 1.3 × 10^{4} and − 1.6 × 10^{6} m^{3}, respectively. These values are similar to those obtained by Rikitake and Sato (1989), who proposed a model of volcanic eruptions triggered by magma reservoir compression due to earthquakes (Nakamura 1971). For instance, using the above estimates of the volume changes of the magma reservoir and cross-sectional area of a vent (*S*), we can estimate the degree of magma head ascent within the vent: we obtain a value of about 2.6 × 10^{6} m^{2} using the following scaling relation (McNutt and Nishimura 2008):

$$ S={10}^{\left(\left(0.46*\mathrm{V}\mathrm{E}\mathrm{I}+0.08\right)-0.52\right)/0.29}\operatorname{} $$

(2)

where the VEI of the 1707 Hoei eruption was 5. Separately, if we assume an ejecta volume (*V*) of about 1 km^{3} for the Hoei eruption, a scaling law between crater diameter (*D*) and ejecta volume (Sato and Taniguchi 1997)

$$ D=0.11{V}^{0.42} $$

(3)

results in a crater diameter of *D* ~ 6.6 × 10^{2} m and a cross-sectional area of *S* ~ 1.4 × 10^{6} m^{2} (approximated as a circle). On the basis of the above results, we estimate the ascent of the magma head within the vent to be on the order of 10^{− 2} − 10^{0} m. These small values seem insufficient to trigger an eruption unless additional effects occur, such as magma mixing.

In addition, the effects of dynamic stress changes, which we do not examine in this study, may promote eruption triggering (Manga and Brodsky 2006). We acknowledge that elastodynamic effects alone could explain eruptions following distant large earthquakes many hundreds of kilometers away, but there is no evidence of elastodynamic effects exceeding the elastostatic effects due to nearby earthquakes. As an example, Ichihara and Brodsky (2006) revealed that rectified diffusion, one of the popular physical mechanisms of elastodynamic effects, causes pressure changes of at most 2 × 10^{− 9} of initial values for seismic waves with typical amplitudes. If this is true, then elastodynamic effects due to rectified diffusion are negligible compared with the elastostatic effects reported in this study. Moreover, it seems difficult to explain the rarity of the 1707 Hoei case in terms of elastodynamic effects compared with elastostatic effects. In Fig. 11 and below, we further discuss volcano responses to earthquake deformation.

### Statistical analysis of seismicity after the 2011 Tohoku earthquake

The 2011 Tohoku earthquake has not yet triggered an eruption at Mt. Fuji, but seismicity levels beneath Mt. Fuji have increased since the event (Enescu et al. 2012; Kumazawa and Ogata 2013). A recent geophysical study (Fujita et al. 2013) modeled an Mw 5.9 earthquake beneath Mt. Fuji on March 15, 2011 as occurring on a strike-slip fault. This event followed the 2011 Tohoku earthquake by 4 days, and many aftershocks occurred. We show earthquakes near Mt. Fuji after the 2011 Tohoku earthquake in Fig. 7. Those post-Tohoku earthquakes and (magmatic) low-frequency earthquakes in 2000–2014 seem to align nearly north–south, i.e., in a different direction to the extended dike plane described in previous sections. Figure 8 illustrates the hypocentral depths of post-Tohoku and low-frequency earthquakes. The depths of the former tend to be slightly shallower than those of the latter.

Whether or not post-Tohoku seismicity reflects magma movement at depth is scientifically interesting. Here, we statistically analyze seismicity in 2011 for the region shown in Fig. 7 using the ETAS (epidemic-type aftershock sequence) model (Ogata 1988). Adopting an ETAS model allows us to evaluate the background seismicity rate *μ* without the effects of aftershocks. A seismological study by Llenos et al. (2009) proposed that changes in *μ* reflect alterations to aseismic stressing rates around earthquake hypocenters. We now apply this concept to seismicity around Mt. Fuji.

The ETAS model is briefly summarized as follows. (1) Every earthquake causes its own aftershocks, obeying the modified Omori law (Omori 1894; Utsu 1961) for temporal decay. (2) Aftershock productivity shows an exponential relationship with earthquake magnitude. (3) Seismicity rate at time *t* is the sum of prior aftershock sequences and background seismicity rate. Thus, the seismicity rate *λ*(*t*) is given by

$$ \lambda (t)=\mu +{\displaystyle \sum_{t_i\le t}}\frac{K\ {e}^{\alpha \left({M}_i-{M}_c\right)}}{{\left(t-{t}_i+c\right)}^p} $$

(4)

where *μ* is the background seismicity rate, *M*
_{
i
} is the magnitude of the *i*th earthquake, *M*
_{
c
} is the minimum earthquake magnitude, and the other parameters (*K*, *c*, *α*, *p*) are constants.

To estimate the model parameters, we use SASeis2006 (Ogata 2006), which estimates the five ETAS parameters (*μ*, *K*, *c*, *α*, *p*) by minimizing the value of the Akaike Information Criterion (Akaike 1974) with a fast likelihood algorithm (Ogata et al. 1993). For the actual parameter estimation, we set *M*
_{
c
} = 1.0 to remove small earthquakes from calculation of the Gutenberg–Richter law (Gutenberg and Richter 1944) for seismic data in this region. We also assume initial parameter values for the estimation algorithm of (*μ*, *K*, *c*, *α*, *p*) = (1, 1, 0.5, 1, 1.5).

Figure 9 shows the temporal evolution of the background seismicity rate *μ* in 2011. The value of *μ* from the Tohoku earthquake (11 March 2011) until the next month (~30 April 2011) is one order of magnitude larger than that of the other periods. This fact suggests notable aseismic stressing around the earthquake hypocenters, since the background seismicity rate *μ* reflects an aseismic stress rate, possibly controlled by rate- and state-dependent friction (Dieterich 1994), as proposed by Llenos et al. (2009). This aseismic stressing and the hypocenter depths (slightly shallower than for low-frequency earthquakes) might reflect magma activity above the reservoir. The nearly north–south alignment, indicated from the hypocenters of post-Tohoku seismicity and low-frequency earthquakes (Fig. 7), suggests that magma ascended slightly along a weak zone, namely the source regions of the post-Tohoku earthquakes, including a Mw 5.9 fault and low-frequency earthquakes, similar to transform-parallel dikes (Gudmundsson 2007). Although it is unclear when the weak zone began to form, this could be a representative magma pathway for past or future activity. Herein, we call it “PPP” (possible pathway plane), similar to the dike plane.

Figure 10 illustrates the normal stress changes perpendicular to the PPP due to the five earthquakes in Fig. 3. For the cases of magma reservoir compression in Fig. 5a, b, e, the PPP is totally clamped and so may not favor the triggering of an eruption (see the previous section). The stress change for the ISTL earthquake (d) has a unique pattern, with unclamping in the northern region and clamping in the southern part. In case (c) (i.e., the 2011 Tohoku earthquake and corresponding magma reservoir dilation), the PPP is totally clamped in a way that differs from the (extended) dike plane. It is probable that dilation of the magma reservoir had a weak positive effect on the ascent process of gas bubbles, but clamping restrained the bubble ascent, which then increased the stress in the area that produced the local earthquakes. The fact that the corresponding Mw 5.9 earthquake was a strike-slip earthquake (Fujita et al. 2013) is not curious; for instance, a geodetic study reported dike-triggered aseismic strike-slip in Tanzania (Himematsu and Furuya 2015).

Why earthquakes now occur along the PPP instead of the (extended) dike plane is an ongoing mystery. Recent geophysical observations have found no evidence for volcanic activity along the (extended) dike plane. One possible hypothesis is that the main pathway of the magma changed from the (extended) dike plane following the 1707 Hoei eruption. Since the 1707 Hoei eruption was the most explosive historic eruption of Mt. Fuji, it might have changed the internal structure and stress state around the volcanic edifice. Another hypothesis is that the path of magma ascent fluctuates on relatively short time scales, and the present pathway is not along the (extended) dike plane. This hypothesis may be supported by the star-like shape of the fissure distribution (e.g., Fig. 2), but what causes the fluctuations is truly unknown. If such fluctuations are controlled by the thermal diffusion system, then their time scales would be far longer than those discussed in this study.

### Implications for volcano response following earthquake-induced deformation

Figure 11 shows the calculated results of elastostatic changes around the magma reservoir and pathways (extended dike plane and PPP). The right half of the figure corresponds to the area of the former model of volcano–earthquake interactions proposed by Walter and Amelung (2007). The case of the 2011 Tohoku earthquake and post-Tohoku seismicity along the PPP of Mt. Fuji fall into this category. The upper left quadrant of the figure corresponds to the hypothesis of Nostro et al. (1998). The case of the 1707 Hoei earthquake and the subsequent Hoei eruption in the deep part of the dike plane (leading to magma mixing) are located in this part of the figure.

In many other cases, the magma reservoir was compressed and the dike was wholly clamped (lower left quadrant in Fig. 11). It seems that the volcanic system was not activated in these cases. This is because the compression of the magma reservoir itself would not be enough to trigger an eruption, as discussed in the previous sections. If other large earthquakes compressed the magma reservoir and unclamped the dike beneath Mt. Fuji, the volcano system might respond acutely, as discussed previously (Nostro et al. 1998). Likewise, if another large tectonic earthquake dilated the magma reservoir by an order of magnitude more than the 2011 Tohoku earthquake and the ISTL earthquake, then the volcano system might possibly respond as proposed in a previous study (Walter and Amelung 2007). However, we cannot identify a tectonic earthquake large enough to produce such dilation. This fact implies that the magmatic system beneath Mt. Fuji is almost insulated from the elastostatic influences of large tectonic earthquakes.

### Effects of other possible local events near Mt. Fuji

As a possible scenario, we examine another local earthquake near Mt. Fuji, along the Fujikawa-kako fault zone (Lin et al. 2013), that is not a large tectonic earthquake. This inland fault zone is located north of the source fault of the Ansei Tokai earthquake and possibly is a northern extension of the plate boundary (the left-hand boundary in Fig. 1). We only consider the southern segment of the fault zone (near the coast) described by a geological study (Lin et al. 2013). Hereinafter, we will refer to this as the FKFZ earthquake. We modeled the FKFZ earthquake as a west-dipping pure reverse fault with a dip angle of 30° and a strike of N180° E. The length and width of the fault are each 15 km, and the slip amount is 5 m. The location of the northern corner of the upper edge is 138.6° E, 35.25° N, and the depth is 1 km. The moment magnitude of this local earthquake is 7.0, with an assumed rigidity of 30 GPa. Figure 12 shows the calculation results.

Around the northern parts of the magma pathways (both the extended dike plane and PPP), the magma reservoir is compressed and the pathways are clamped, similar to the effects of the 1703 Kanto earthquake on the extended dike plane. As discussed in the previous section, this does not favor eruption triggering. In contrast, the calculation results for the southern parts of the magma pathways (both the extended dike plane and PPP) are qualitatively similar to the case of the Tohoku earthquake. Namely, the magma reservoir is dilated while the extended dike plane is unclamped and the PPP is clamped. We note that the values of the elastostatic effects are somewhat larger than the case of the 2011 Tohoku earthquake. This local FKFZ earthquake may have the power to trigger an eruption around the southern parts of the magma pathways, following the model of Walter and Amelung (2007).

Another possible seismic event that could affect Mt. Fuji is one or more M7 slow slip events downdip of the source fault of the Tokai earthquake (Ozawa et al. 2016). Since the location of the slow slip events is near the Tokai earthquake, the effects would be qualitatively similar to, but orders of magnitude smaller than, the Tokai earthquake (case (e) in the previous sections).