Introduction

Phreatic eruptions are driven by subsurface hydrothermal water at shallow level beneath volcanoes and geothermal zones. Although these eruptions are usually small, with erupted volumes of <0.01 km3, they occur frequently in volcanic settings and have occasionally been fatal because they may occur with no warning1,2,3,4,5. This abrupt eruption style, whereby the eruption starts at a shallow depth beneath the volcano and possesses no clear signature of magma intrusion from depth, makes the forecasting of phreatic eruptions difficult. Although an increase in shallow seismicity has been observed days to months before some phreatic eruptions2,3,4, this scenario is not a generality, as this increase in seismicity often returns to its background level without an eruption. Furthermore, phreatic eruptions commonly occur with no precursory increase in seismicity, such as the 2018 eruption at Moto-Shirane Volcano1,6.

Near-vent observations have shown that a short-term precursor occurs before the onset of phreatic eruptions. Tiltmeter and seismometer observations near the eruptive centre (usually <2 km) have revealed that ground deformation with tremor occurs several minutes before the potential onset of phreatic eruptions1,2,6,7,8,9,10,11. The onset time of deformation and tremor coincides, thereby indicating that both changes are generated by the same subsurface process. Such a precursory deformation with tremor (DT) episode is believed to be related to subsurface fluid intrusion; therefore, DT episodes may be a key to better understanding the processes leading to a phreatic eruption. However, DT episodes also occur with no eruptive activity12,13,14, such that more details on DT episodes at the time of a phreatic eruption and those that occurred with no associated eruption need to be obtained to elucidate the potential relationship between DT episodes and phreatic activity.

A small phreatic eruption occurred on 19 April 2018 at Iwo-Yama (or Io-Yama), Kirishima Volcanic Complex, Japan15,16. Historical records indicate that Iwo-Yama has experienced repeated phreatic eruptions17; however, it had been in a dormant state since the 1990s, with no signs of fumarole activity at the surface. Fumarole activity resumed in December 2015, with gas exsolution at ~80 °C in the crater for the first time in 12 years. The temperature soon increased to boiling point (~95 °C) at the altitude of the Iwo-Yama and the fumarolic area expanded intermittently15. DT episodes have occurred 13 times around Iwo-Yama since August 2014, including at the time of the 2018 phreatic eruption, which emitted ~1500 t volcanic ash15. Here we analyse geophysical data that were recorded around Iwo-Yama, and highlight the importance of groundwater flow, which is inferred from electric-field changes, during both the DT episodes and phreatic eruption.

Observations

The Japan Meteorological Agency (JMA) has continuously operated a tiltmeter and seismometer at the KKN-NE site, which is located 2200 m NNE of the Iwo-Yama (Fig. 1a, b), since March 2013, with the sensors installed in the bottom of a 95 m deep borehole. A seismometer was also installed at the KKN-NE site. We have continuously operated tiltmeters at the KVO sites, which are located 1000 m W of Iwo-Yama (Fig. 1a, b) since May 2010. The first DT episode was detected on 20 August 2014. A magnetotelluric (MT) site, which records the horizontal electric- and geomagnetic-field time series at the surface, has been operated continuously since May 201118 for the purpose of monitoring the subsurface electrical resistivity structure. The voltage difference (in mV) in the N–S and E–W directions are recorded at a 32 Hz sample rate using non-polarisable electrodes (Pb-PbCl) and wires with a Metronix ADU07e data logger. The electric-field time series (in mV km−1) is calculated by dividing the recorded voltage differences by the dipole length (N–S: 40 m; E–W: 35 m). The long time series of these observations allow us to compare the DT episode that occurred at the time of the phreatic eruption to those that occurred with no associated eruption. Two additional tiltmeters (IWO-SW and KREB) and a visual camera were established (Fig. 1a, b) in December 2016. The camera, which is located 100 m south of the eruption vent, recorded the entire phreatic sequence (19 April 2018) at a 6 s sampling interval. To the best of our knowledge, this is the first video recording of a phreatic eruption sequence that captures the onset of the eruption and how it evolves to explosive activity at the surface. The unique dataset of proximal geophysical observations and real-time video footage provides us with the opportunity to examine the entire process of a phreatic eruption. The data acquisition system is summarised in the “Methods” section.

Fig. 1: Observation site locations and geophysical data recorded during typical deformation with tremor (DT) episodes.
figure 1

a Topographic map of the Kirishima Volcanic Complex, Japan (100-m contour interval). Yellow and green arrows indicate the averaged downward tilt vectors during a DT episode at the KKN-NE and KVO sites, respectively. The KKN-NE data are exaggerated five times compared with the data from the KVO site. b Close-up view of the rectangular area in (a). c Typical tilt, seismogram, and electric-field records for DT episodes. Site names are shown to the left. The possible onset of the event is indicated by the vertical dotted lines. The northward and eastward directions are positive in the electric field. Note that the events shown here did not generate a phreatic eruption.

Results

Data features during DT episodes with no eruption

Figure 1 shows the geophysical data that were recorded during the DT episodes that were not associated with an eruption. The DT episodes at Iwo-Yama usually continue for several minutes (Fig. 1c), as has been observed at other volcanoes. Thirteen DT episodes, including the 19 April 2018 episode during the phreatic eruption, are plotted over time in Fig. 2. The downward tilt directions at the KKN-NE site always point to approximately N105°E ± 3° (Fig. 2a), and those at the KVO site point to approximately S7°W ± 11°, with the exception of a very small amplitude event on 2 September 2015 (Fig. 2a). Figure 1c shows that the electric-field changes occur simultaneously with the DT episode. We note that the amplitudes of the electric-field changes are much larger than the electromagnetic induction amplitudes (“Methods” section and Supplementary Fig. S1). We also note that the temporal increase (approximately several minutes) in temperature around the electrodes is excluded as a cause of the electric-field changes because there is no known fumarolic zone within 400 m of the MT site. The most plausible mechanism of this electric-field change is its electrokinetic origin, which is caused by groundwater flow in porous media19,20,21. The electric-field change provides us with information about local subsurface groundwater flow, which cannot be estimated from other geophysical observations. For example, cyclic electric-field changes have been observed around geysers, with the corresponding groundwater movement estimated from these changes22,23. The direction of the electric-field change via the electrokinetic mechanism is usually the opposite of that produced via groundwater flow19,20,21. The direction of the electric-field changes at the MT site is basically NW (i.e., SE groundwater flow), although variations are observed (Figs. 1c, 2), despite the relatively stable tilt directions at KKN-NE and KVO.

Fig. 2: Temporal evolution of DT episodes and tectonic earthquake activity around Iwo-Yama Volcano.
figure 2

Fumarole activity resumed on Iwo-Yama during 2–14 December 2015 following a 12-year hiatus. a Tilt and electric-field observations associated with the DT episodes. The tilt directions (yellow: KKN-NE; green: KVO; blue: KREB; cyan: IWO-SW) and directions of the electric-field changes (squares) at the MT site are shown. The signals associated with the 19 April 2018 phreatic eruption are highlighted in red. The symbol size is normalised by the data amplitude at the sites. No symbols mean missing records due to equipment issues. b Ratio between the tilt at KKN-NE and electric-field change amplitudes at MT. c Histogram of shallow (<3 km below sea level) volcano tectonic (VT) earthquake activity around Iwo-Yama. The black and white bars indicate the earthquakes that occurred within a 1- and 2-km radius of the Iwo-Yama crater, respectively.

Data features during a phreatic eruption

Figure 3 shows the geophysical data that were recorded during the phreatic eruption on 19 April 2018. The eruption was clearly accompanied by a precursory DT episode that began at 15:34:30 Japan Standard Time (JST) and continued for approximately 3 min based on the recorded tilt signals at KKN-NE and KVO (Fig. 3 and Supplementary Fig. S2). Only the tiltmeter that was 250 m WSW of the eruption vent (IWO-SW) began to exhibit unique changes after the first three minutes. This is the first time that a near-vent tiltmeter exhibited substantial changes a few minutes after the onset of a DT episode (Figs. 1, 3 and Supplementary Fig. S2).

Fig. 3: Data recorded during the 19 April 2018 phreatic eruption.
figure 3

a Tilt, seismogram, and electric-field records. The site names and their distances to the eruption vent are noted to the left. The four vertical lines with numbers correspond to the camera snapshots in (b). Note that substantial tilt changes were recorded at approximately time ii at the near-vent IWO-SW site, whereas these changes were not recorded at the far-vent KKN-NE site and the other two sites (Supplementary Fig. S2). The NS electric-field changes reach a maximum before iv, which is when the explosion occurred. b Camera snapshots that were taken from the site 100 m to the south of the eruption vent (Fig. 1).

The electric field exhibits a typical change during the first three minutes of the eruption, when no anomalous surface phenomena occurred. However, both the northward electric field and the fumaroles on Iwo-Yama started to increase at 15:38:10 JST (ii in Fig. 3a), approximately one minute before the onset of steam effusion. Steam effusion began 100 m to the south of the Iwo-Yama crater at 15:39:20 JST (iii in Fig. 3a). The northward electric field increased substantially (~10 mV/km) as the steam effusion intensified. Note that the MT site is located NNE of Iwo-Yama and the eruption vent (Fig. 1a, b), and the increase in the northward electric field suggests groundwater flow to the vent. The explosion occurred at 15:44:15 JST (iv in Fig. 3a), approximately five minutes after the onset of steam effusion and consisted of a cock’s tail plume that emitted rocks and blackish ash (Fig. 3b). Therefore, to the best of our knowledge, this study is the first to report electric-field changes associated with both DT episodes and the subsequent phreatic eruption. The plume became steam dominant again after the explosion, and the northward electric field gradually returned to its background level. Both of explosion and DT episode have not occurred again to date (June 2022).

Discussion

A comparison of the near-vent and far-vent tilt records provides inferences on the dynamics of the phreatic eruption. Both the far-vent and near-vent tiltmeters recorded gradual changes during the first three minutes, but only the near-vent tiltmeter (IWO-SW) recorded a substantial change during the minute before steam effusion (Fig. 3a and Supplementary Fig. S2). These changes imply that at least two dynamic processes were occurring before the eruption: one is a relatively deep intrusion that induced the tilt change at both the far-vent and near-vent tiltmeters, and the other is a delayed shallow intrusion(s) that only induced near-vent tilt changes. These subsurface pressure changes are not isotropic but have the shape of either a dyke or sill, as the direction of the tilt vectors is not consistent with the directions of either the eruption vent or Iwo-Yama crater. Another important observation is that no additional tilt changes occurred before the explosion, whereas the electric field underwent substantial changes. The observed electric-field changes before the explosion cannot be explained by the electric field generated by the poroelastic response24, as there were no additional tilt changes, which indicates that there were also no additional pressure changes.

Here we interpret all of the DT episodes to be caused by the intrusion of hydrothermal fluids, as there is no evidence of a magmatic contribution. We first investigated the DT episodes with no coincident eruption (January 2013 to June 2018). Figure 2c shows that the number of shallow tectonic earthquakes around Iwo-Yama increased since January 2014, with their hypocentres being concentrated around Iwo-Yama since June 2014; the first DT episode occurred on 20 August 2014. Shallow volcano tectonic earthquakes at Iwo-Yama have been interpreted as being stimulated by the pore-pressure build-up beneath a low-permeability clay layer and the subsequent reduction in effective normal stress16. The increase in seismicity before the first DT episode suggests that a localised high pore-pressure environment is necessary for the occurrence of DT episodes. Figure 2b shows an increase in the amplitude ratio (electric field versus tilt at KKN-NE) from June 2015 to the present. The tilt amplitude is proportional to the volume of the hydrothermal intrusion when it is assumed that the shape, location, and elastic properties of the background structure do not change25,26. Therefore, this amplitude ratio may reflect the efficiency in inducing groundwater flow per unit volume of hydrothermal intrusion. An increase in the amplitude ratio likely reflects the enhanced efficiency in inducing groundwater flow after the first DT episode in June 2015. This interpretation is consistent with the fact that fumarole activity resumed at the surface in December 2015 after being absent for 12 years.

We propose the following model of the DT episodes and associated groundwater flow to explain both the cases with and without a coincident phreatic eruption (Fig. 4) based on the observed changes in the tilt and electric-field records. The background structure is based on the MT-derived electrical resistivity structure16,18. A low-permeability clay layer that partially seals the underlying hot (>200 °C) hydrothermal water and an overlying cold low-salinity aquifer (10–40 °C) that extends from the surface to 100–200 m depth are present beneath Iwo-Yama16,18. The existence of a near-surface cold aquifer is supported by the existence of crater lakes and creeks at the surface (Fig. 1b). A small amount of hot hydrothermal water was continuously supplied from the base of the clay-rich layer through fractures and mixed with near-surface groundwater before the eruption, thereby maintaining a fumarolic zone (~95 °C) at the surface (Fig. 4a)16. Here we interpret the electric-field changes to be generated by the movement of near-surface, cold, low-salinity groundwater (Fig. 4). High-salinity hydrothermal water flow through fractures at depth is not considered to contribute to the observed electric-field changes due to the decrease in the electrokinetic effect with increasing fluid salinity27,28,29,30. The contribution from near-surface cold groundwater flow is supported by the fact that a substatial electric-field increase (~10 mV/km) occurred approximately three minutes after the onset of precursory DT activity (Fig. 3).

Fig. 4: Schematic model of the 19 April 2018 phreatic eruption.
figure 4

The numbers (i to iv) correspond to the time shown in Fig. 3. The map view and cross section for each snapshot of the model indicate the geometry of inflation sources (red rectangles) and a deflation Mogi source (cyan circle). a Before the onset of the eruption. A small amount of hot hydrothermal fluids is supplied from the base of the clay-rich layer through fractures. b Early stage of the DT episode. Hot hydrothermal fluids intrude the clay-rich layer and extend northward to shallow depth. An influx of groundwater flow occurs near the fumarolic zone, while groundwater flows away from the extended sill. c Before the onset of steam effusion. An additional hydrothermal fluid (brine) intrudes in the shallow aquifer along the direction of the eruption vent, and induces the strong influx of groundwater to the fumarolic zone. d Time up to the explosion. The boiling becomes vigorous beneath the vent, and a large amount of groundwater flows to the vent. Explosive phreatic eruptions occur via a direct liquid-to-vapour phase transition of hot hydrothermal fluids after the groundwater flow is reduced and no longer able to cool the hydrothermal intrusion.

We interpret the DT episodes to be caused by the intrusion of hydrothermal fluids from the bottom of the clay-rich layer via pre-existing fractures in the clay-rich layer (Fig. 4a), with this process generating numerous small fractures that cause the tremor. The intrusion further promotes boiling at the base of the near-surface aquifer beneath the fumarolic zone, and subsequently promotes the influx of cold groundwater to compensate for the vaporised water, which in turn causes the observed increase in the northward electric field at the MT site (Figs. 1c, 3a and 4b). When an additional sill extends northward, eastward groundwater movement may also be induced near the MT site by the pressure increase around the sill, thereby causing a westward electric-field change (Figs. 1c, 3a, and 4b).

We modelled the temporal evolution of the tilt changes during the 2018 phreatic eruption (“Methods” section). The tilt vectors during the first three minutes, which are similar to the DT episodes with no eruptions, are explained by the above scenario (Fig. 4b and Supplementary Fig. S3), which considers a deeper sub-vertical dyke (240–570 m beneath the erupted vent; Dyke 1) and a shallower sill (200–230 m beneath the erupted vent; Dyke 2). The estimated volumes of the dyke and sill are 1.14 × 105 and 5.05 × 104 m3, respectively. This deeper sub-vertical dyke, which is a common feature during all of the DT episodes, simultaneously causes the tilt changes at both of the far- and near-vent tiltmeters. The location of the sill is consistent with a model that was derived from an interferometric synthetic aperture radar analysis of long-term (2014–2016, 2016–2017, and 2017–2018) inflation31.

The near-vent tiltmeter (IWO-SW) begins to exhibit unique changes one minute before the onset of steam effusion (Fig. 3a). Another dyke (8.42 × 103 m3; Dyke 3) that is along the direction of the erupted vent at 80–150 m depth beneath the erupted vent, which corresponds to the depth of the near-surface aquifer, is therefore necessary to produce this tiltmeter response (Fig. 4c and Supplementary Fig. 4). This additional dyke substantially promotes both the boiling of the aquifer beneath the vent and the influx of the surrounding cold groundwater (northward electric-field increase). Steam effusion possibly began when the boiling zone reached the surface (Fig. 4c).

The northward electric field continued to increase after the onset of steam effusion, which indicates that both near-surface aquifer boiling and groundwater flow became vigorous. The explosion occurred approximately five minutes after the onset of steam effusion (10 min after the onset of the DT episode). Volcanic explosions are usually considered to occur when the pore-fluid pressure becomes greater than the overlying rock strength. However, the absence of tilt changes suggests that there were no substantial pressure changes before the explosion, which means that the vaporised water effectively escaped to the air before the explosion. Conversely, the electric field exhibited substantial changes before the explosion. The temporal changes in the northward electric field (Fig. 3a) indicate that the explosion occurred when the near-surface groundwater flow to the vent began to decrease after reaching its maximum. We suggest that the final trigger of the explosion was the depletion of this near-surface groundwater flow around the vent. Such a condition may have accelerated the direct liquid-to-gas phase transition of the hot (>200 °C) hydrothermal fluid intrusion, which led to the phreatic explosion (Fig. 4d). The explosion was not accompanied by subsequent deflation (Fig. 3a), which suggests that the volume of hydrothermal fluids in the intrusions did not change in response to the explosion. This is consistent with the fact that strong steam effusion continued after the explosion32.

Our proposed model of the 2018 phreatic eruption explains the tilt vector observations and induced electric-field changes. The results suggest that the DT episode was due to the shallow intrusion(s) of hot (>200 °C) hydrothermal fluids. Thirteen DT episodes were recorded at Iwo-Yama, among which twelve were not associated with an eruption. The DT episodes that were not associated with an eruption are interpreted to be failed phreatic eruptions that were inhibited by shallow groundwater flow. The influx of groundwater flow and its vaporisation usually cools hot hydrothermal fluids from depth, and potentially acts as a buffer to inhibit phreatic eruptions. However, substantial changes in both the near-vent tilt and the electric field are induced when hot hydrothermal fluids intrude in the near-surface aquifer beneath the active vent. Explosive phreatic eruptions occur when cold groundwater is depleted and its flow is reduced around the vent.

In summary, our speculated sequence for the occurrence of a phreatic eruption is as follows: (1) pore-pressure build-up first occurs beneath the clay-rich layer16, as observed by the shallow seismicity; (2) the intrusion of hot hydrothermal fluids and subsequent groundwater flow then cools the intrusion, as observed by the DT episodes and moderate electric-field changes; (3) there is an intrusion of hot hydrothermal fluids in the near-surface aquifer and a substantial influx of cold groundwater, as observed by the substantial near-vent tilt and electric-field changes; and (4) there is a direct liquid-to-vapour phase transition due to the depletion of groundwater flow, as observed by the decay in the electric-field changes. A temporal increase in seismicity may not be observed before phreatic eruptions if the pore pressure has already reached a critical threshold. Therefore, the observation of processes (2) to (4) is also important for understanding and forecasting phreatic eruptions. The presence and flow of near-surface groundwater may be key in controlling the occurrence of phreatic eruptions that can be monitored using electric-field measurements.

Methods

Observation system

The MT site is operated by the Kyushu and Tokyo Universities18. The 32 Hz time series of the electric field (in mV km−1) is down-sampled to 1 Hz by averaging, and a high-pass filter (<5000 s) is then applied. JMA operates the two geophysical sites (KKN-NE and IWO-SW) on Iwo-Yama. A tiltmeter (Mitsutoyo JTS-3BH) and seismometer (Mitsutoyo V224) were installed at the bottom of a 95 m-deep borehole at the KKN-NE site; the resolutions are >0.01 micro-radian and >0.001 micro-m s−1, respectively. The tilt data are low-pass filtered at >30 s and digitised at 1 Hz, and the seismic data are sampled at 100 Hz. A tiltmeter (Pinnacle Denali) was installed at the bottom of a 15 m-deep borehole in December 2016 at the IWO-SW site; its resolution is <0.001 micro-radian. The tilt data are low-pass filtered at >30 s and digitised at 1 Hz. Kyushu, Kagoshima, and Tokyo universities operated two additional tilt sites (KVO and KREB). Applied Geomechanics 701-2 A dual-axis tiltmeters with a 0.1 micro-radian resolution were installed in the horizontal tunnel (corresponding depth of ~5 m) at the KVO site and in the floor of the cabin at the KREB site. The KREB tilt records were initially sampled at 100 Hz and have been down-sampled to 1 Hz by averaging. The KVO tilt records were sampled at 1 Hz. Here, we further applied a 10 s moving window with 90% overlap to the 1 Hz data at the KVO and KREB sites. All of the time-series data used in this study are GPS-synchronised. The data acquisition is summarised in Table 1.

Table 1 Summary of the data acquisition.

Electric-field time series data

Earth’s electric field contains an induction component that captures the temporal changes in the natural electromagnetic field. The induction component in the MT method is used to estimate the subsurface electrical resistivity structure. However, such component is treated as noise in this study. Therefore, we removed the induction component in the electric-field time series data. The induction component of the electric field is expressed in the frequency domain as:

$$\left(\begin{array}{c}{{{E}}}_{{{x}}}\,(f)\\ {{{E}}}_{{{y}}}\,(f)\end{array}\right)=\left(\begin{array}{cc}{{{Z}}}_{{{xx}}}\,(f) & {{{Z}}}_{{{xy}}}\,(f)\\ {{{Z}}}_{{{yx}}}\,(f) & {{{Z}}}_{{{yy}}}\,(f)\end{array}\right)\left(\begin{array}{c}{{{B}}}_{{{x}}}\,(f)\\ {{{B}}}_{{{y}}}\,(f)\end{array}\right),$$
(1)

where Ex and Ey are the northward and eastward electric fields, respectively, and Bx and By are the northward and eastward magnetic fields, respectively. The long-term MT measurements allow four high-quality components of the impedance tensor Z to be estimated. We used the 1-Hz sampling geomagnetic field data obtained at Kakioka geomagnetic observatory33 for the Bx and By components. We did not use the measured geomagnetic field at the MT station, as the geomagnetic field data at the MT station exhibited large fluctuations during the DT episodes, which may have been due to the inclination of the induction coils. The low-frequency (<0.1 Hz) induction component of the electric field during a given DT episode was predicted by using the estimated impedance tensor Z and the Kakioka 1 Hz sampling geomagnetic field at the time of the DT episode, and then subtracted from the original time series data. Figures 13 show the electric-field changes after applying this procedure. Note that the procedure does not substantially affect the results because the electric-field variations at the time of the DT episode are larger than the induction effect (Supplementary Fig. S1).

Order estimation of the pore-pressure gradient that induced groundwater flow

We calculated the order of the dynamic pore-pressure gradient that caused the cold groundwater flow and corresponding electric-field change via the Helmholtz–Smoluchowski (H–S) equation:

$$\frac{\nabla V}{\nabla P}=\frac{{{{\varepsilon }}}_{0}{{{\varepsilon }}}_{{{r}}}\zeta }{{{{{{{\rm{\sigma }}}}}}}_{{{f}}}\mu },$$
(2)

which relates the voltage (V: in mV) and pressure (P: in Pa) changes that are induced in porous media for ideal situations, such as laboratory experiments19,28,34. σf, ε0, εr, ζ, and μ are the electrical conductivity of the fluid (in S m−1), dielectric constant of a vacuum, relative dielectric constant, zeta potential of the rock–water system (in V), and dynamic viscosity of the fluid (in Pa s), respectively. Note that Eq. (2) assumes laminar flow at a lower frequency range in fully saturated porous media, such that the surface electrical conductivity is negligible19,35,36. Substitution of the approximate amplitude at the MT site (V = 10 mV km−1), approximate zeta potential (ζ = −20 mV) of rock samples within 5 km of Iwo-Yama that were measured in the laboratory in a 0.001 mol L-1 NaCl solution34, whose resistivity corresponds to ~0.01 S m−1 at 15 °C, ε0 = 8.9 × 10–12, εr = 80, μ = 1.3 × 10–3 Pa∙s, and approximate water resistivity (σf = 0.01 S m−1 at 15 °C), which was measured at nearby (~200 m) ponds37, into Eq. (2) yields P = ~10 Pa m−1. Note that this value is the estimation at the MT site, which is 500 m from the eruption vent.

The relationship between the pore-pressure gradient P and water flow j (m s−1) are represented by Darcy’s law:

$$j=-\frac{\kappa }{\mu }\nabla P,$$
(3)

where κ is the permeability of the near-surface aquifer. If we assume radial groundwater flux to the vent from all directions, and consider that the distance from the vent and the MT site, and the thickness of the near-surface aquifer, are ~500 and ~100 m16, respectively, then the total flux to the vent is calculated as ~2 × 109 κ m3 s−1. The total cold groundwater influx to the vent is calculated as 20 kg s−1 before the explosion when we assume κ = 10–11 m2.

Tilt modelling

The temporal evolution of the tilt changes at the time of the 2018 phreatic eruption was modelled for two time periods. One period is from 15:34:35 to 15:37:00 JST, which corresponds to when the tilt changes ended at the far-vent site (KKN-NE). The near-vent tiltmeters then began to exhibit substantial changes at 15:37:00 JST. The other period is from 15:34:35 to 15:39:15 JST, which corresponds to the time when steam effusion began. We modelled multiple finite rectangular tensile cracks based on a dislocation model by Okada26 and a spherical deflation source25 to explain the observed tilt changes during the two time periods. The surface displacement in the Okada (1992) model is represented analytically by eight parameters: the spatial location of the crack centre (XO, YO, and ZO); horizontal length (L) and vertical width (W) of the crack; strike of the crack (A) relative to north; dip angle (D) relative to the ground surface; and dislocation (U) of the crack. The spherical source (Mogi source) is represented by four parameters: its spatial location (XM, YM, and ZM) and volume change (V). Topography was considered in the model. We assumed a Lamé’s constant of λ = μ and searched for the best-fit parameter combination for tensile cracks (XO, YO, ZO, L, W, A, D, and U for each crack) and a deflation source (XM, YM, ZM, and V) via the least-squares method.