Introduction

Geofluids (e.g. aqueous fluids and magma) in the Earth’s interior play an important role in various geodynamic processes; they affect the chemical reactions and physical properties of the crust and mantle. For example, a reduction in rock strength due to the introduction of aqueous fluids can induce seismicity1,2, plate motion3, and subduction4. Aqueous fluids expelled from subducting plates can react with the mantle rocks, lowering the rock’s melting temperature and producing magma5. These aqueous fluids and magmas can rise and result in arc volcanism5, seismicity and crustal deformation6, and continental crust growth over a long period7.

Previous studies employed various observational methods, such as seismic or magnetotelluric (MT) tomography, to elucidate the distribution of geofluids in the Earth’s interior. Several parameters, including seismic P- and S-wave velocities (VP and VS, respectively), seismic attenuation, and electrical conductivity (σ)8,9,10,11, have been used to detect geofluids. Slow seismic velocities with large VP/VS ratios and attenuations, or high σ, may be interpreted as an indication of geofluids12,13.

However, determining the presence and distribution of geofluids, particularly in the crust, is difficult. For example, the VP/VS ratios of the major rock types that constitute the crust (VP/VS = 1.5–2.0) exhibit enough diversity to explain the observed variations for scenarios where fluids are absent14. Furthermore, owing to the relatively high compressibility of aqueous fluids, the VP/VS ratio of rocks containing these fluids may decrease (compared to that of dry, intact rocks) depending on the amount of fluid, the geometrical shape of the fluid (as expressed by the aspect ratio), and temperature–pressure conditions14.

Contrastingly, electrical conductivity is more responsive to the presence of geofluids, and major crustal rocks show relatively low electrical conductivity in the absence of fluids (<10−2 S/m)14 compared to the 1 S/m observed conductivity for arcs and continental crust. Therefore, higher σ values suggest the presence of geofluids. However, determining the exact amount and connectivity of the fluids based solely on the electrical conductivities is difficult9,14.

As either seismic velocity or electrical conductivity alone is insufficient for constraining the distribution of geofluids, previous studies conducted analyses that combined the two observations10,15. A few studies also conducted quantitative analyses for the combined observations, referred to as ‘joint inversion’ or ‘simultaneous analysis’8,14,16,17. In this study, we applied a recently developed inversion method to simultaneously analyse seismic velocity and electrical conductivity18 using high-resolution tomographic results12,13 to quantitatively map fluid volume and determine the geometrical parameters of the geofluids in the Northeast Japan arc.

Results and discussion

Seicmic velocity and electrical conductivity structures

The Northeast Japan arc is geodynamically active, with geological features such as earthquakes, hot springs, volcanoes, and crustal deformation. The study area corresponds to a trench-arc segment that encompasses the hypocentre and ruptured area of ‘the 2011 off the Pacific coast of Tohoku Earthquake’ (Fig. 1). Since 2000, more than 50,000 earthquakes have been recorded in the study area, including the 2008 Iwate-Miyagi Nairiku earthquake [hereinafter referred to as IMN2008, with an MJMA of 7.2 (MW of 6.9)]. Several Quaternary volcanoes are situated in the area. Of these, Kurikoma Volcano, which consists primarily of andesitic lavas and pyroclastic rocks (ranging from basaltic andesite to dacite), occupies the central part and has experienced frequent eruptions since ~0.5 Ma, with the latest phreatomagmatic eruption occurring in 194419. The 2011 Tohoku Earthquake caused the volcanic area to subside by up to 10 cm, creating elliptical depressions, with the horizontal dimensions being 15–20 km20. Notably, more than 90 hot springs have existed in the area since before IMN2008, half of which have temperatures exceeding 80 °C21 (Fig. 1). Thus, this area is suitable for studying the relationship between earthquakes, geofluids, and crustal deformation through geofluid mapping.

Fig. 1: Index maps of the study area.
Fig. 1: Index maps of the study area.
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a Study area in the Northeast Japan arc subducted by the Pacific plate, with the relative velocity being 73 mm/yr38. The figure illustrates the hypocentre and ruptured area of ‘the 2011 off the Pacific coast of Tohoku Earthquake’ (solid star and white broken line, respectively)39 and the volcanic front (red dashed line). b The seismic and magnetotelluric (MT) stations (181 red circles and 110 blue circles, respectively) considered for this study and the inversion grid nodes of 20367 points (small sky-blue circles) for which the P- and S-wave seismic velocities and electrical conductivity were obtained (Supplementary Fig. 1). c Epicentres of 51,358 earthquakes that occurred during 2000–2018 (small green dots have depths ≧ 6 km and light-blue dots have depths <6 km); the open star indicates the hypocentre of the 2008 Iwate-Miyagi Nairiku earthquake (IMN2008) of the magnitude Mj = 7.240. The five boxes represent the fault planes associated with IMN200841. Quaternary volcanoes are shown using red triangles42, and hot springs21 are shown using circles.

The seismic structure of the study area was determined based on the tomography results obtained in a previous study13, where manually selected P- and S-wave arrival time data from nearly 20,000 earthquakes that occurred during 2006–2012 were used. The grid was set at intervals of 0.0625° (~7 km) in the N-S and E-W directions and depths of 0.0, 2.5, 5.0, 7.5, 10.0, 15.0, 20.0, 25.0, 30.0, 35.0, 40.0, 65.0, 90.0, 120.0, 150.0, and 180.0 km13(Supplementary Fig. 1d). The fitting criterion for the seismic model is the minimisation of the travel time residual by revising the three-dimensional seismic velocity structure. The uncertainty of the velocity was estimated to be up to 3%.

The electrical conductivity structure of the study area was determined based on a three-dimensional (3D) inversion22 of a wide-band MT dataset that covers a range of 0.3–2000 s, consisting of 110 stations spaced 5 km apart. This MT dataset is a compilation of pre-existing23,24 and newly obtained datasets (Methods, Supplementary Figs. 1–3, and Supplementary Data). The inversion process utilised the full tensor impedances measured at eight representative periods, logarithmically spanning 0.25–800 s, with a 10% error floor. The model consists of a 1-km horizontal grid mesh in the core zone and an outer coarse mesh that includes the surrounding oceans. The vertical meshes are coarser logarithmically at greater depths. The initial model consists of a uniform Earth with an electrical conductivity of 0.01 S/m and a fixed ocean resistivity of 4 S/m. The final model achieves a root mean square (RMS) value of 1.53 (Methods).

The tomography results at 10 km (Fig. 2a, b) and other depths (Supplementary Figs. 4–7) show that VP/VS and log10 σ exhibit significant variations mostly from 1.6 to 1.9 and −4 to 1, respectively, with distinct localised features, e.g. a high electrical conductivity region was noted just beneath the IMN2008 hypocentre (Fig. 2b). While VP and VS show a high correlation (with a correlation coefficient of 0.956, Fig. 2c), VP, VS, VP/VS, and log10 σ show lower correlation coefficient values (Fig. 2d, e); this trend is particularly noted between seismic velocity and electrical conductivity (0.075, 0.077, and 0.030 for log10 σ  vs. VP, VS, and VP/VS, respectively). The low values suggest variable lithology-geofluid combinations with different geofluid parameters (i.e. liquid fraction and geometry); therefore, a simultaneous analysis of seismic velocity and electrical conductivity is necessary14.

Fig. 2: Seismic velocity and electrical conductivity in the study area.
Fig. 2: Seismic velocity and electrical conductivity in the study area.
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Horizontal distribution of (a) VP/VS and (b) electrical conductivity (log10 σ) at 10-km depth. The white star denotes the epicentre of the 2008 Iwate-Miyagi Nairiku earthquake (IMN2008), and the small red triangles represent the Quaternary volcanoes, with the easternmost corresponding to the Kurikoma Volcano on the volcanic front (Fig. 1). Correlations between (c) VP and VS, (d) VP and VP/VS, and (e) VS and log10 σ. Data points with estimated geofluid volumes greater than 3% (Supplementary Data) are shown in red.

Although VP and VS are relatively sensitive to lithological variations and σ is more sensitive to liquid phase variations, they are related through the common parameters, particularly the amount of liquid phase. For instance, in Fig. 2e, data points in red with high σ are estimated to involve geofluids of more than 3 vol%, while some with low σ may also involve the appreciable amounts of geofluids. Discriminating such conditions and quantifying the amount of geofluids require VP and VS data, in addition to σ.

In this study, we use a recently developed inversion method to quantitatively estimate the lithology, geofluid type, and relevant fraction and geometrical parameters in relation to the aspect ratio and connectivity of the geofluid phase, based on a simultaneous analysis of VP, VS, and σ14,18. The values for VP, VS, and σ come from the final models of seismic tomography and MT inversions separately and then the simultaneous analysis is being done. The optimal solution was obtained probabilistically by applying a marginalisation technique within the Bayesian statistical framework18 (Methods).

Quantitative mapping of geofluids

The inversion analysis first integrates the probability by grid search over the entire range of the three fluid parameters (ϕ, α, and αEC) for each of the 234 combinations (78 lithologies × three types of geofluids [aqueous fluid, basaltic magma, and andesitic magma, Methods]). Based on the integrated probability, the most probable lithology-geofluid combination is determined, and then the most probable values of ϕ, α, and αEC are calculated18. This method does not necessarily minimise the residual between the data and the model for each data point. However, the resulting residuals are mostly within the data uncertainties (Supplementary Figs. 4c, 5c, and 7c), demonstrating the model’s ability to reproduce the observed data effectively. Moreover, the model shows an average probability of 14% (Supplementary Fig. 8), which is well constrained compared to the probability of a uniform distribution (i.e., 1/234 × 100 = 0.43%).

Figure 3a, b shows the most probable distributions of the aqueous fluid and andesitic and parental basaltic magmas in the study area (Supplementary Data). The probabilities (p) of the solutions in the Bayesian framework (Fig. 3c) are relatively high and sharply focused in most areas (>10%), including the deep part around the Moho (Supplementary Fig. 8). In contrast, the p value for the middle crust–shallow part of the western part is relatively low (<10%, Supplementary Fig. 8), and the probability distribution was not sharply focused (e.g., the ‘4%’ case, Fig. 3c). However, stable solutions are obtained to specify the type of geofluid even for the ‘4%’ case (Fig. 3c) and these solutions are well constrained compared to the aforementioned probability of 0.43% for a uniform distribution. In addition, inversion results of a synthetic data set (Supplementary Fig. 9) demonstrate that the three types of geofluids can be reasonably identified and quantified under the crust and uppermost mantle conditions.

Fig. 3: Geofluid map and the probability.
Fig. 3: Geofluid map and the probability.
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Distribution of aqueous fluid and basaltic and andesitic magmas for 140.45–141.15°E, 38.72–39.15°N, at depths ranging from 1–40 km. Views from the (a) top and (b) south for saline aqueous fluid (blue, upper row) and basaltic (green, mid row) and andesitic (red, bottom row) magmas. In a, b the dark-coloured regions (with 10 vol% geofluids) are surrounded by pale transparent regions (3 vol%). The probabilities for the optimal solutions in a, b are shown in (c), presenting three examples (at three grid nodes) of the probability distribution, including the 234 lithology-geofluid combinations (78 lithologies × three types of geofluids). The 78 lithologies range from felsic to ultramafic in composition (horizontal axis of the three example diagrams). The three types of geofluids represent the aqueous fluid (blue) and andesitic (red), and basaltic (green) magmas.

The upper rows of Fig. 3a, b show the distribution of aqueous fluid. The most prominent feature is a fluid reservoir with a large amount of aqueous fluid (~10 vol%), distributed at a depth of 10–20 km below the hypocentre of IMN2008 (Fig. 4). We observe an inverse correlation between earthquake frequency and the aqueous fluid volume. The maximum number of earthquakes occurred at a depth of 5–6 km; the number of earthquakes decreases with depth, whereas the amount of fluid increases (Fig. 4). Notably, earthquakes rarely occurred in the fluid reservoir.

Fig. 4: Vertical profiles of the geofluid parameters.
Fig. 4: Vertical profiles of the geofluid parameters.
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a The average volume profile of liquid phases beneath the IMN2008 hypocentre (Fig. 3). The range in volume at each depth is shown as an error bar. The frequency of earthquakes shown in Fig. 1 is the highest (~10,000) in the area directly above the fluid reservoir with log10 ϕ ∼ −1. In b line (2) represents the calculated pressure profile of the flowing aqueous fluid (see Methods), with reference to the hydrostatic profiles shown by lines (1) and (3). The excess fluid pressure, ΔP, over the lithostatic pressure is denoted by the red line.

In this reservoir located under the hypocentre of IMN2008, the fluid volume, ϕ, decreases from ~10% (at a depth of 20 km) to 0.1% (at a depth of ~30 km, Fig. 4a). For the depth of 20–30 km, the average value of αEC (a threshold fluid fraction to achieve liquid-phase connection and efficient electrical conduction) is 0.1–0.01%, lower than the fraction of aqueous fluid; this suggests that the fluid is connected throughout the depth range (Supplementary Fig. 10). Magma exists just below this depth, and the fluid is rooted in the magma (Fig. 4a) that spreads along the Moho (Fig. 5 and Supplementary Fig. 11 for depth slices).

Fig. 5: Geofluid map showing the connection between magmas, aqueous fluids, earthquakes, and volcanoes.
Fig. 5: Geofluid map showing the connection between magmas, aqueous fluids, earthquakes, and volcanoes.
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Distribution of geofluids (with >3 vol%) and the across-arc E-W structure between 140.45°–141.15°E and 38.95°–39.05°N, showing the relationships between basaltic (green) and andesitic (red) magmas, aqueous fluid (pale blue), earthquakes, and volcanoes.

Notably, the magma does not spread homogenously along the Moho and exhibited distinct spatial and compositional variations. Basaltic magma is distributed mostly in the uppermost mantle and on the western side of the study area (under the volcanic zone, Figs. 1 and 3b). It is barely detected in the shallow crust and was not observed directly under the volcano. Contrastingly, andesitic magma is identified in the shallow part, including the areas directly beneath the volcanoes, as well as being widely distributed near the Moho (under the fore-arc zone, Figs. 1 and 3b). We observe a magma-depleted region (gap) near the Moho (Fig. 5) between the basaltic magma distributed mainly on the western side and the andesitic magma distributed in the east, suggesting that the magma that originally spread near the Moho ascended above this gap to feed the volcanoes (Fig. 5).

Magma, fluid, and earthquake connection

In Northeast Japan, the addition of fluid from the subducted Pacific Plate to the mantle wedge initiates mantle melting at depths between 70 and 90 km5, which produces basaltic magma that ascends towards the Moho. This magma loses buoyancy, owing to the changes in lithology and rock density at the Moho (Supplementary Fig. 12), spreading laterally25 to the fore- and back-arc zones. Although the back-arc zone is outside our study area, a low-velocity anomaly in this zone has been reported previously13, which could be attributed to a westward migration of magma. During this lateral migration, basaltic magma cools and undergoes crystallisation, producing the andesitic magma in the fore-arc zone (Fig. 5). Solidification occurs in the marginal part of the magma-rich region near the Moho, and the aqueous fluid components dissolved in the magma are gradually expelled, forming a fluid reservoir under the IMN2008 hypocentre (Fig. 5).

During the gradual fluid expulsion that occurs at a depth of 32 km (Point A, Fig. 4b), the pressures of the aqueous fluid, magma, and host rock are nearly equal in the quasi-equilibrium state. However, at the surface (Point B), the fluid pressure is close to atmospheric or slightly higher, as indicated by the artesian springs. The fluid between points A and B is interconnected, as previously described, and the fluid pressure decreases with the vertical distance from Point A, influenced by fluid velocity and permeability, until reaching the atmospheric pressure at Point B. The numerical solution satisfying these conditions (see Methods) revealed large excess pressure from the central–shallow regions of the crust. Particularly, the depth range where the fluid pressure exceeds the lithostatic pressure by >200 MPa corresponds to the seismic activity zone (Fig. 4b), implying that the aqueous fluid induced earthquakes in the region (including the IMN2008). At the depth of the fluid reservoir, in addition to the lower excess pressure of the fluid, the higher temperature promotes plastic deformation, thereby suppressing seismicity.

In addition to the high seismicity area surrounding the IMN2008 hypocentre, we observe another high seismicity area in the shallower crust (<6 km in depth) south of Kurikoma Volcano (Fig. 1); no significant fluid reservoir existed in the subsurface of this region. Earthquakes in the southern area occurred along multiple fault planes (roughly running N-S, with E-W compressional displacement) associated with IMN2008 and its fault plane (Fig. 1).

Notably, in the eastern part of the study area (on the trench side of the volcanic front, Fig. 1), including the IMN2008 region and the abovementioned high seismicity areas, the magma is present near the Moho (Figs. 3b, 5, and Supplementary Fig. 11). The presence of magma deep in the fore-arc zone suggests that the location of the volcanic front was not solely determined by the magma distribution in the mantle wedge. The mantle processes that occur at depths of 50–200 km, including the dehydration of a subducted plate, distribution of temperature and melting within the mantle wedge, and melt migration dynamics, may control and define the positions of volcanic fronts26,27. However, our results suggest that the magma supplied from the mantle may spread near the Moho, with the location of the volcanic front remaining unknown. Although the mantle processes described in this section contribute to the formation of the volcanic front, our results suggest that the processes that occur around the Moho and at shallower depths, e.g. horizontal migration and crystallisation along the Moho and the subsequent ascent of magma (Fig. 5) and structural control associated with reactivation of the numerous Miocene faults in the volcanic zone under across-arc compression since ~3.5 Ma28, play an important role in the formation of volcanic fronts in the Northeast Japan arc.

Geofluid mapping of our study requires a high-resolution dataset including seismic velocities and electrical conductivity. Such datasets have been relatively rare; however, once the dataset is constructed, our study can provide a reliable framework for determining the volume, geometrical parameters, and pressure of aqueous fluid phase and different magma types in complex tectonic settings, thereby fuelling detailed studies on geofluid dynamics, crustal deformation, and earthquakes, in the fields of geodynamics, geoengineering, and disaster mitigation sciences.

Methods

Magnetotelluric (MT) inversion

The MT method29 is an electromagnetic geophysical method that measures the electrical resistivity of the Earth’s sub-surface. The method uses natural electromagnetic signals that originate from volcanic lightning (period <1 s) and magnetospheric storms (period >1 s). We measured the electromagnetic fields in a wide period band (between 3 ms and 2000 s) and estimated the impedance tensors as the Earth’s response functions for each period and location. Then, we carried out 3D inversion modelling22 to obtain a 3D conductivity structure that could explain the observed impedance tensor with minimum model features. In this study, the inversion started with a uniform earth model (σ = 0.01 S/m), while assuming a fixed distribution of the surrounding ocean. The procedure was iterated owing to the nonlinearity of the problem. We used the MT inversion code of ref. 22. We inverted full components of impedance tensor with 10% error floor. The initial model and the prior model are a 100 Ωm uniform earth with 0.25 Ωm ocean. The RMS of the initial model is 10.09. The inversions were iterated with changing the trade-off parameter ‘lambda’, and the smallest RMS model was found as the best model. Then, the best model was used as the initial and prior model to obtain the second-generation best model. This procedure was repeated for five-times and the RMS of 1.53 was reached.

Constraints and prior information considered in this study

The volcanoes in the study area, represented by Kurikoma Volcano, are located approximately along the volcanic front of the Northeast Japan Arc (Fig. 1). The volcanic rocks contain phenocrysts consisting of plagioclase, clinopyroxene, orthopyroxene, opaque minerals, and olivine19. Some major element concentrations (but not complete sets), such as SiO2 and FeO/MgO in bulk rock compositions, have been reported in the literature19; the lavas from Kurikoma Volcano are mostly andesitic and exhibit a continuous trend from basaltic andesite to dacite (with the SiO2 content ranging from 56 wt% to 66 wt%), suggesting a fractionation trend from basaltic magma. Based on these observations, we use the reported compositions of IW17 and HM0130,31, which are representative of the relatively undifferentiated basaltic and andesitic magmas along the volcanic front of the Northeast Japan Arc, with the SiO2 contents being 51.4 and 60.9 wt% and FeO/MgO being 1.3 and 1.6, respectively; both IW17 and HM01 are estimated to contain 5 wt% H2O30,31.

The saline aqueous fluids in the study area represent the deep-seated fluids derived from either the magmas32 and/or the subducting plate beneath the Japan arcs33. Based on the composition of the spring water with the highest salinity observed in the study area, we estimate that the NaCl concentration in the aqueous fluid is 2 wt%. We use the abovementioned compositions for the inversion calculation (for the basaltic and andesitic magmas and aqueous fluid), to map the geofluids with probability.

The geothermal gradient and depth–temperature relation used in this study are based on the crustal heat flow observations presented in the literature34 and the numerical simulations accounting for mantle convection using the structure shown in a previous work14. Compared with that in the fore-arc zone, the heat flow in the volcanic zone to the west of the volcanic front is discontinuously higher34, where active volcanoes occur, and hot magmas ascend to the surface (Fig. 1). Based on these observations, we assume that the temperatures are 1100 and 900 °C for the basaltic and andesitic magmas, respectively, to calculate the physical properties of the magmas in the volcanic zone.

We consider 78 rock types14, which cover the observed seismic velocities sufficiently, with relatively no bias, to obtain the most probable solutions for lithology, geofluid type (aqueous fluid, basaltic magma, andesitic magma), ϕ (geofluid volume), α (aspect ratio of pore filled by geofluids)35, and αEC (threshold fraction required to achieve liquid-phase connection)14.

Probabilistic inversion method

To obtain the most probable solution, in addition to the values of VP, VS, and σ, we utilise the near-surface observations (heat flow and the compositions of lavas and spring waters that were described in the previous section) as prior knowledge. We calculate the probabilities of the presence of these geofluids, including saline aqueous fluid, andesitic magma that appears on the surface, and basaltic magma that is potentially seated at different depths as a parent magma, determining combinations with 78 lithologies and variable geometrical parameters of the geofluid phase as below.

In this study, we use a new probabilistic inversion method in the Bayesian statistical framework18 to estimate the following parameters. [1] The 78 possible lithologies are labelled using discrete variables iΛ (=1, 2, · · ·, 78); these encompassed felsic–ultramafic rocks (including sedimentary, metamorphic, and igneous rocks14). [2] The geofluid types (aqueous fluid or basaltic magma or andesitic magma) are labelled as discrete variables iβ (=1, 2, 3). [3] The geofluid amount (volume fraction, ϕ, ranging from 10−7 to 1) is determined as a continuous variable. [4] We determine the parameters related to fluid geometry (aspect ratio, α, from 10−5 to 1) and the critical fluid fraction related to connectivity (αEC,  from 10−5 to 1) as continuous variables,14 based on the VP, VS, and σ data obtained from the tomography results.

The detailed calculation procedure used in this study is outlined in a previous study18. The method uses a marginalisation technique that first estimates the values of iΛ and iβ and then quantifies the values of ϕ, α, and αEC. This method is robust and can be reliably used for lithology and geofluid-type determination, as indicated by the inversion tests conducted in a previous study using synthetic datasets18. By using the forward model14, we calculate the P-wave velocity (fP), S-wave velocity (fS) and electrical conductivity (fσ) for a given set of iΛ, iβ, ϕ, α, αEC, geofluid composition (major-element compositions of the magmas and salinities of aqueous fluids), pressure, and temperature at each grid node (Fig. 1b). The geofluid compositions are constrained based on the observations of the lavas and hot-spring waters, and the pressure and temperature are constrained by the surface heat flow and numerical simulations for a coupled system of plate subduction, induced mantle convection in the wedge, and heat conduction in the overriding crust14. For each grid node, all combinations of iΛ, iβ, ϕ, α, and αEC were examined based on the evaluation function, E0 [see Eq. (1)] as the negative logarithm of the joint posterior probability, p, of the unknown parameters (iΛ, iβ, ϕ, α, and αEC), based on the tomography results.

$${E}_{0}(i{\scriptstyle{\varLambda}} ,\,{i}_{\beta }, {{\phi}},{{{\rm{\alpha }}}},{{{\rm{\alpha}}}}_{{{\rm{EC}}}}) \, \equiv \, {-}{{{\mathrm{ln}}}} \, p \, ({i}_{\varLambda },\,{i}_{\beta }, {\phi},{{{\rm{\alpha }}}},{\alpha }_{{EC}}|{V}_{P},{V}_{S},\sigma ) \\ = \, \{{V}_{P}{-}{f}_{P}({i}_{\varLambda },{i}_{\beta }, {\phi},{{{\rm{\alpha }}}},{{{\rm{\alpha }}}}_{{{\rm{EC}}}})\}^{2}/({2}\,{s_{P}}^{2})\\ +\{{V}_{S}{-}{f}_{S}({i}_{\varLambda },\,{i}_{\beta }, {\phi},{{{\rm{\alpha }}}},{{{{\rm{\alpha }}}}_{{{\rm{EC}}}}})\}^{2}/({2}\,{s_{S}}^{2}) \\ +\{{\log }_{10}{{{\rm{\sigma }}}}{-}{\log }_{10} \, f _\sigma ({i}_{\varLambda },{i}_{\beta }, {\phi},{{{\rm{\alpha }}}},{{{\rm{\alpha }}}}_{{{\rm{EC}}}})\}^{2}/({2}\,{s_{\sigma}}^{2})+{{{\rm{C}}}}$$
(1)

where sP, sS, and sσ represent the standard deviations of the uncertainties in VP, VS, and log10 σ, respectively, and C is a constant that is independent of the unknown parameters. In this study, 2sP and 2sS represent 3.0% of the observed VP and VS, according to Okada et al.13, and the value of 2sσ is 0.5 for the log10 σ, according to Ogawa et al.12.

First, the most probable combination of iΛ and iβ was selected to minimise Σϕ, α, αEC{E0 (iΛ, iβ, ϕ, α, αEC)}, which corresponded to the maximisation of the marginalised posterior probability of iΛ and iβ, based on the tomography results, p(iΛ, iβ | ϕ, α, αEC, VP, VS, σ). Note that the summation symbol Σx,y {F(x, y, z)} indicates the summation of F(x, y, z) over all possible combinations of x and y.

Then, using the selected iΛ† and iβ† values, the most probable values of ϕ, α, and αEC were calculated by minimising E0 (iΛ†, iβ†, ϕ, α, αEC), which corresponded to the maximisation of the posterior probability of ϕ, α, and αEC, given iΛ†, iβ†, VP, VS, and σ, p(ϕ, α, αEC | iΛ†, iβ†, VP, VS, σ). The probability of lithology-geofluid combinations in the main text corresponds to the abovementioned marginalised posterior probability, p(iΛ, iβ | ϕ, α, αEC, VP, VS, σ), calculated using the following equation:

$${\Sigma }_{\phi, \, {{{\rm{\alpha }}}}, \, {{{{\rm{\alpha }}}}}_{{{{\rm{EC}}}}}}\left[\exp \left\{{-}{E}_{0}({i}_{\varLambda },{i}_{\beta },\phi ,{{{\rm{\alpha }}}},{{{{\rm{\alpha }}}}}_{{{{\rm{EC}}}}})\right\}\right]/ \\ {\Sigma }_{{i}_{\varLambda }, \, {i}_{\beta }, \, \phi , \, {{{\rm{\alpha }}}}, \, {{{{\rm{\alpha }}}}}_{{{{\rm{EC}}}}}}\left[\exp \left\{{-}{E}_{0}({i}_{\varLambda },{i}_{\beta },\phi ,{{{\rm{\alpha }}}},{{{{\rm{\alpha }}}}}_{{{{\rm{EC}}}}})\right\}\right].$$
(2)

In this inversion analysis of VP, VS, and σ, spatial smoothing was not applied to avoid duplicated smoothing, in addition to the spatial smoothing already applied in the individual seismic and MT inversions. Nonetheless, it is likely that some important structures, such as low-velocity anomalies, tend to be under-recovered, leading to an underestimation of geofluid fraction ϕ. In this respect, the estimated ϕ of the geofluid-abundant regions in Figs. 3–5 is likely to represent a minimum value, providing strong evidence for the presence of geofluids.

Fluid flow and pressure

In the area around the hypocentre of IMN2008, ϕ exceeds 0.1% for the entire depth range (from the top near the surface to the root of the aqueous fluids at 32 km depth); this includes the fluid reservoir (ϕ ~ 10%) at a depth of ~10–20 km (Supplementary Fig. 10). For this depth range, the average value of αEC is less than 0.1% (Supplementary Fig. 10), suggesting that the fluid is connected throughout the depth profile. When the connected fluid is stationary, the fluid pressure gradient follows the hydrostatic pressure gradient, and the two cases (starting from the surface and the root of the aqueous fluid) provided the lower and upper bounds of the hydrostatic pressure [lines (1) and (3) in Fig. 4b, respectively]. When the connected fluid flows, the pressure gradient, ∇P, deviates from the hydrostatic pressure gradient in accordance with the flow velocity (Vaf) and permeability (kϕ), which can be expressed as follows36:

$${V}_{{af}}={-}{k}_{\phi}\nabla P/{\phi}_{{{\rm{\eta }}}},{{{\rm{and}}}}\,{k}_{\phi}={R}^{2} {\phi}^{3}/B$$
(3)

where the viscosity of the aqueous fluid, η, is set at 10−3 Pa s 37, the mineral grain size of the permeable rock is set as 10−3 m, and the constant B is set as 103, 36. This equation involves the viscous force between the fluid flow and the rocks, which decreases the fluid pressure along the flow. By using the average ϕ at each depth shown in Fig. 4a and assuming a constant fluid flux, ϕVaf, over the entire depth range, we numerically solved the equation for the depth variation of P, to satisfy both the atmospheric pressure at the surface (Point B, Fig. 4b) and the lithostatic pressure at the depth of 32 km (Point A; as illustrated by line (2) in Fig. 4b). The calculated ϕVaf value is 2.1 × 10−10 m/s per unit area, comparable to the average discharge rate of groundwater in the study area21 (~10−10 m/s, based on the discharge rate of 10−2 km3/year over an area of ~3000 km2). If the fluid flux is not constant, e.g. the aqueous fluid reservoir with ϕ ~ 10% supplied the fluid to the shallow depth (without any influx from the deeper part), the depth profile of P has more similarities with line (3), indicating more excess pressure at shallow depths (<10 km). In any case, emptying the aqueous fluid reservoir (with the estimated fluid volume being ~3000 km3) would require ~0.3 million years, based on the outflux estimated in this study. Therefore, the fluid reservoir is not rapidly tapped but leaks weakly.