### Inversion algorithm

We conducted 3-D inversions to infer the resistivity structure corresponding to the ACTIVE responses; for these inversions, we used the finite element (FE) method with an unstructured tetrahedral mesh. In this section, we describe essential parts of our inversion method.

In the forward part, we solve the vector Helmholtz equation in terms of the magnetic vector potential *A*,

$$ {\mathbf{\nabla }} \times \left( {{\mathbf{\nabla }} \times \varvec{A}} \right) + i\omega \mu \hat{\sigma }\varvec{A} = \mu \varvec{i}_{\text{s}} , $$

(1)

by the edge-based FE method (e.g., Yoshimura and Oshiman 2002). Here, \( \omega \) is the angular frequency, \( \mu \) is the magnetic permeability, and \( \hat{\sigma }\left( { \equiv \sigma + i\omega \varepsilon} \right) \) is the complex conductivity, where \( \sigma \) is the electrical conductivity and ε is the electric permittivity; \( \mu \) and ε are assumed to be free-space values throughout this paper. \( \varvec{i}_{\text{s}} \) denotes the source electric current density. The electric and magnetic fields, \( \varvec{E} \) and \( \varvec{B} \), respectively, are linked to \( \varvec{A} \) by the relationships \( \varvec{E} = - i\omega \varvec{A} \) and \( \varvec{B} = \nabla \times \varvec{A} \) (see Yoshimura and Oshiman 2002 for the reason why the scalar potential \( \phi \) is omitted). Equation (1) is first discretized by the FE method with first-order tetrahedral elements and then solved by the weighted residuals method. For the source term in Eq. (1), we adopt a kind of Heaviside function, where \( \varvec{i}_{\text{s}} \) has a constant value only on a straight line within an element (e.g., Ansari and Farquharson 2014). To include the actual topography, we adopt an unstructured tetrahedral mesh generated by Gmsh freeware (http://gmsh.info/).

In the inverse part, we minimize the objective function,

$$ \Phi \left( {\mathbf{m}} \right) = \Phi_{\text{d}} \left( {\mathbf{m}} \right) + \lambda \Phi_{\text{m}} \left( {\mathbf{m}} \right), $$

(2)

$$ \Phi_{d} \left( {\mathbf{m}} \right) = \frac{1}{2}\left( {\varvec{F}\left( {\mathbf{m}} \right) - \varvec{d}} \right)^{T} \varvec{B}_{\text{d}} \left( {\varvec{F}\left( {\mathbf{m}} \right) - \varvec{d}} \right), $$

(3)

$$ \Phi_{\text{m}} \left( {\mathbf{m}} \right) = \frac{1}{2}\left( {{\mathbf{m}} - {\mathbf{m}}_{\text{ref}} } \right)^{T} \varvec{B}_{\text{m}} \left( {{\mathbf{m}} - {\mathbf{m}}_{\text{ref}} } \right), $$

(4)

where \( \lambda \) is a trade-off parameter and \( \varvec{d} \) is a data vector consisting of amplitudes and phases of ACTIVE responses. \( \varvec{F}\left( {\mathbf{m}} \right) \) is the vector resulting from the forward modeling; \( \varvec{B}_{\text{d}} \) is a diagonal weighting matrix consisting of the reciprocals of the squared standard errors; \( {\mathbf{m}} \) is the model vector, defined as \( m_{i} = \log_{10} \rho_{i} \) in this paper, where \( \rho_{i} \) is the resistivity of the *i*-th model block, and \( {\mathbf{m}}_{\text{ref}} \) is the reference model. For the roughening matrix \( \varvec{B}_{\text{m}} \), we adopt \( \varvec{B}_{\text{m}} = \varvec{R}^{T} \varvec{R} \) following Usui et al. (2017). The *i*-th component of the product of \( \varvec{R} \) and **m** is given by,

$$ \left[ {\varvec{R}{\mathbf{m}}} \right]_{i} = \sum\limits_{j}^{{N_{\text{if}} }} {\left( {m_{i} - m_{j} } \right),\varvec{ }} $$

(5)

where \( N_{\text{if}} \) is the number of model blocks neighboring the *i*-th model block. Note that \( \varvec{R} \) is a kind of Laplacian operator for an unstructured tetrahedral mesh (e.g., see the analogous form of \( R_{2} \) in Eq. (2) of Constable et al. 1987).

Equation (2) is minimized by using the Gauss–Newton scheme with a data-space approach (e.g., Siripunvaraporn et al. 2005). For the selection of \( \lambda \), a cooling strategy (e.g., Schwarzbach and Haber 2013) is adopted, in which \( \lambda \) is decreased by a factor of \( 10^{1/3} \) when \( {\text{nRMS}} = \sqrt {2\Phi_{d} \left( {\mathbf{m}} \right)/N_{\text{d}} } \), where \( N_{\text{d}} \) is the length of \( \varvec{d} \), does not decrease by 10% from the previous iteration step.

### Inversions of the ACTIVE datasets from August 2014 to August 2015

We conducted two inversions of ACTIVE datasets, one of the dataset obtained in August 2014, before MEP, and the other dataset obtained in August 2015, after MEP. Because the peak frequencies of temporal variations of the ACTIVE response were similar from November 2014 onward (Fig. 1b), we can infer the temporal variations in the resistivity structure between the period before and that after the magmatic eruptions began from the difference between August 2014 and August 2015. Datasets from four receivers were available for August 2014, but for August 2015, datasets from only three receivers were available. Therefore, we first inverted the four receiver datasets for August 2014 to obtain model \( {\mathbf{m}}_{0} \), and then we inferred model \( {\mathbf{m}}_{1} \) by inverting the three receiver datasets for August 2015. The relationship between \( {\mathbf{m}}_{0} \) and \( {\mathbf{m}}_{1} \) is expressed as,

$$ {\mathbf{m}}_{1} = {\mathbf{m}}_{0} + d{\mathbf{m}}, $$

where \( d{\mathbf{m}} \) corresponds to the temporal variation in the resistivity structure between August 2014 and August 2015. For the \( {\mathbf{m}}_{0} \) inversion, we adopted a modified version of the 3-D resistivity structure obtained by the AMT survey in 2004–2005 (Kanda et al. 2015) as the initial model (see Additional file 1: Fig. S1 and Additional file 1: Fig. S2 for our tetrahedral mesh and the initial model, respectively) and a homogeneous 100 Ωm structure as the reference model in Eq. (4). For the \( {\mathbf{m}}_{1} \) inversion, \( {\mathbf{m}}_{0} \) was used for both the initial and reference models. In both the \( {\mathbf{m}}_{0} \) and \( {\mathbf{m}}_{1} \) inversions, \( \varvec{B}_{\text{d}} \) consisted of the real measurement errors with error floors of 1% and \( \arcsin \left( {0.01} \right)*180/\pi = 0.573 \) degrees for the amplitude and phase, respectively.

Because the available datasets were not adequate in our 3-D inversions, we limited the model space for the \( {\mathbf{m}}_{0} \) inversion vertically to elevations higher than 0.0 m and laterally to a 3 km × 3 km square centered on the first Nakadake crater; outside of this zone, all model blocks were integrated into a single model block corresponding to a homogeneous background medium, which was also determined by the \( {\mathbf{m}}_{0} \) inversion. In the \( {\mathbf{m}}_{1} \) inversion, the background resistivity of \( {\mathbf{m}}_{1} \) was fixed to that of \( {\mathbf{m}}_{0} \), because only three receiver datasets were available.