Dispersion curves are commonly obtained from array measurements of microtremors using the Spatial Auto Correlation (SPAC) method (e.g., Aki 1957; Okada 2003; Asten 2006) or the frequency–wavenumber (*f*–*k*) method (Capon 1969; Lacoss et al. 1969). We used the Extended Spatial Auto Correlation (ESPAC) method (Ling and Okada 1993; Okada 2003) because previous studies have demonstrated that this method provides more reliable estimates of dispersion curves than the *f*–*k* method (e.g., Ohori et al. 2002; Foti et al. 2011). The ESPAC method defines the real parts of cross-coherence as SPAC coefficients and expresses them theoretically using the zero-order Bessel function of the first kind, as follows:

$$\rho \left( {f,\Delta x} \right) = J_{0} \left( {\frac{2\pi f}{V\left( f \right)}\Delta x} \right) ,$$

(1)

where \(\rho\) is the SPAC coefficient, \(f\) frequency, \(\Delta x\) receiver spacing, \(V\left( f \right)\) phase velocity, and \(J_{0}\) the zero-order Bessel function of the first kind.

To estimate dispersion curves, we computed root-mean-square errors (RMSEs) between the observed SPAC coefficient and the Bessel function as defined in Eq. (1) (Additional file 1: Fig. S1c). Dispersion curves at each frequency were then obtained by extracting the data point with the lowest RMSE (dots in Additional file 1: Fig. S1c). The frequency range of the dispersion curve was defined by the following relationship between wavelength and receiver spacing.

$$2\Delta x_{ \text{min} } \, < \,\lambda \, < \,2\Delta x_{ \text{max} } ,$$

(2)

where \(\lambda = \frac{V\left( f \right)}{f}\) is wavelength, \(\Delta x_{ \text{min} }\) the minimum receiver spacing, and \(\Delta x_{ \text{max} }\) the maximum receiver spacing. We manually picked phase velocities considered to be a fundamental mode of Rayleigh waves within the frequency range.

We performed inversion analysis of the resultant dispersion curve to determine the subsurface S-wave velocity structure (Additional file 1: Fig. S1d). The theoretical dispersion curve was calculated by the compound matrix method (Saito and Kabasawa 1993) based on the assumption that the subsurface under the array is in the form of a horizontal multi-layered structure. As the initial model for inversion, we constructed S-wave velocity models by converting phase velocity and wavelength into S-wave velocity and depth by the Ballard method (Ballard 1964) as follows:

$$V_{s} \, = \,1.1\, \times \,V\left( f \right) ,$$

(3)

$$D = \frac{\lambda }{3} ,$$

(4)

where \(V_{s}\) is S-wave velocity and \(D\) depth. We used 5 layers when the maximum depth computed from Eq. (4) is shallower than 50 m. However, when the maximum depth is deeper than 50 m, we used more layers. Thicknesses of each layer were defined so as they increase constantly with depth. Then, we minimized the residual between the observed and theoretical dispersion curves of the assumed S-wave velocity structure using five iterations of the non-linear least-squares method (e.g., Xia et al. 1999). During inversion, we searched only the S-wave velocity for those layers; P-wave velocities and densities were computed from S-wave velocities using the empirical equations of Kitsunezaki et al. (1990) and Ludwig et al. (1970), respectively.

Because the size of the array and the underground structure differ from each observation point, the frequency range of estimated phase velocities (i.e., investigation depth) varied. To clarify the spatial geological structure, we constructed a 3D S-wave velocity structure by interpolating among the inverted S-wave velocity structures at each observation point. For 3D interpolation, we used the Natural Neighbor Method (Sibson 1981), based on Voronoi tessellation of a discrete set of spatial points. This interpolation method has advantages over simpler methods of interpolation, such as nearest-neighbor interpolation, because it provides a smoother approximation of the underlying real function. To perform the interpolation, we first inserted the 3D S-wave velocity data into a volume of 3D grid cells (\(446\, \times \,304\, \times \,101\) cells in the north–south, east–west, and vertical directions, respectively), which covered our study area. Cell dimensions were 10 m (horizontal) and 1 m (vertical), so the total dimensions of the volume were ~ 5 km north–south, ~ 3 km east–west, and 100 m deep, sufficient to cover the ~ 2-km-diameter displaced area. We did not extrapolate the velocity data to the edges of the 3D volume, and fixed S-wave velocities at 0 m (100 m/s) and 100 m depth (300 m/s). These S-wave velocities at boundaries were derived from *N*-values obtained from boreholes in the study area. To clarify the resolution of our model, we performed checkerboard tests (Additional file 1: Fig. S2). Because the input velocity model is recovered in the checkerboard tests, we can interpret the subsurface structures identified in the 3D model.