Several studies have linked changes in seismic velocity to groundwater recharge by rain precipitation (e.g., Gassenmeier et al. 2015; Sens-Schönfelder and Wegler 2006). When surface water from precipitation replenishes groundwater, we expect decrease in seismic velocity reflecting pore pressure increase due to (a) immediate loading in undrained condition (impermeable), and (b) pore pressure diffusion (Talwani 1997). To confirm the effect of precipitation on changes in seismic velocity, we applied two-step analyses.

In the first step, the time delay between precipitation and seismic velocity change is estimated by cross-correlating the two time series. We identify the locations where velocity changes occurred after precipitations, indicating the influence of pore pressure change due to groundwater recharge. Because the quasi-annual period of seismic velocity change could be influenced by other environmental factors (e.g., atmospheric pressure and sea level), we apply a band-pass filter in order to clearly distinguish rainfall from sea level and atmospheric pressure. Therefore, in the first step, we focus on shorter period fluctuations associated with rain precipitation.

In the second step, we focus on locations where precipitation influence on seismic velocity is clearly identified from the first step. We model pore pressure change based on a diffusion mechanism by groundwater load and compare that with the longer period of seismic velocity change to estimate diffusion rate in deep lithology. Although the observed response is mostly a coupled mechanism (i.e. undrained response and pore pressure diffusion), the effect of pore pressure diffusion may be dominant in the later time, as the pore pressure increase due to diffusion occurs once the immediate loading has dissipated (Talwani 1997). The longer period velocity variation may include the influence of sea level and atmospheric pressure effects, but the seismic velocity variation we used here does not include strong annual features associated with sea level and atmospheric pressure. We summarized the flow of these two-step analyses in the flowchart (Fig. 2).

### Step 1: investigation of the rainfall infiltration

To determine an optimal frequency band to clearly distinguish precipitation influences from other environmental factors, we first investigated the power spectra of seismic velocity changes, precipitation, sea-level changes, and atmospheric pressure changes (Additional file 1: Figure S1a). Whereas the power spectrum of seismic velocity changes decreased toward a frequency of 0.1 cycle/day, the spectra of precipitation, sea-level, and atmospheric pressure change showed similar peaks at 0.0018–0.0036 cycle/day, a frequency band close to the annual cycle (Additional file 1: Figure S1b). The similarity of these three peaks meant that the long-term estimated seismic velocity changes could be affected not only by precipitation, but also by sea-level and atmospheric pressure changes.

We excluded frequencies below 0.0036 cycle/day to remove the annual seasonal influence of sea-level and atmospheric pressure changes, and we excluded frequencies above 0.05 cycle/day to eliminate the neap and spring tides of sea-level change and the decreasing spectrum of seismic velocity change. We then searched for the frequency band where precipitation could best be distinguished from sea-level change and atmospheric pressure change, as indicated by weak correlations between precipitation and the other two variables. We applied a band-pass filter for periods between 20 and 137 days (0.05 to 0.0073 cycle/day; Additional file 1: Figures S1c, d) and sought minima in the correlation coefficients between precipitation and sea-level change and between precipitation and atmospheric pressure change, based on the data for all stations. The correlation coefficients were based on the Pearson correlation,

$$\rho \left(\text{A},\text{B}\right)= \frac{\text{cov}\left(\text{A},\text{B}\right)}{\sigma _\text{A} \sigma _\text{B}},$$

(6)

where \(\text{cov}(\text{A},\text{B})\) is the covariance of time series A and B, and \(\sigma _\text{A}\) and \(\sigma _\text{B}\) are the standard deviations of time series A and B.

Figure 3 shows an example of the unfiltered and filtered data for one seismic station (N.YSHH; red dot in Fig. 1f). Seismic velocity changes in Fig. 3a represent the averaged velocity change within a 40-km radius. The unfiltered data are coloured by the stretching correlation coefficient, averaged for station pairs used to estimate the velocity change. In general, the mean correlation coefficient for station pairs used in this study is above 0.5, and the value is even higher in the period when precipitation is relatively high (e.g., August–November). This indicates the daily seismic velocity change in the study area is stable. Figure 3b–d shows the time series of precipitation, sea level, and atmospheric pressure, respectively. Because the Pearson correlation value between rainfall and sea level is very small (Fig. 3e), we can use band-pass filtering to separate the imprint of precipitation and sea level, as well as precipitation and atmospheric pressure. The correlation coefficients between band-pass filtered precipitation and sea-level and between precipitation and atmospheric pressure change, respectively, are shown in Fig. 4 for all stations. The small correlation coefficients indicate that rainfall is distinguishable from sea-level and atmospheric pressure changes.

To further analyse the dependence of seismic velocity changes on rainfall, we applied various time shifts to the rainfall record and evaluated the resulting cross-correlations with seismic velocity changes, as depicted in Fig. 5. Under the assumption that seismic velocity changes are triggered by precipitation after a time lag, we restricted ourselves to positive time lags (i.e. velocity variation after precipitation) and determined the time shift that produced the largest Pearson correlation coefficient.

Although we focus on the shorter cycle (shorter than annual period) in order to clarify the relationship between precipitation and velocity change, it is difficult to distinguish the effects of (a) undrained due to loading and (b) diffusion. Thus, in the next investigation, we evaluate the pore pressure diffusion via modelling.

### Step 2: investigation of pore pressure diffusion

To calculate the pore pressure change, we use the poroelastic model developed by Talwani et al. (2007). The pore pressure due to diffusion can be described as:

$${P}_{k}=\sum_{k=1}^{n}\delta {p}_{k}erfc\left[\frac{r}{({4c\delta {t}_{k})}^{1/2}}\right]$$

(7)

where \(\delta {p}_{k}\) is the water level change, \(r\) is the depth from the surface, \(c\) indicates the hydraulic diffusion, and \(\delta {t}_{k}\) indicates the time increment from the starting time *k* to \(n\), and *erfc* denotes error complementary function. Although Talwani et al. (2007) also proposed equation for the pore pressure changes due to undrain loading, the effect is smaller than the diffusion in longer period. Here, we consider the contribution of precipitation from 365 days in the past, thus current pore pressure change is calculated by using the summation of pore pressure change from the previous 365 days. We defined water level change from 2015 to 2017 as the deviation from the average precipitation over 2014.

To evaluate the longer period variation, we applied a moving average with 130 days windows for seismic velocity change without band-pass filtering in the first step. By comparing seismic velocity changes with the pore pressure changes computed based on Eq. (7), we estimate the optimum hydraulic diffusion at each station. However, in the calculation of the pore pressure changes, it is difficult to constrain the dependence of the hydraulic diffusion with depth because the relative values of the both parameters in \(r/\sqrt{c}\) is sensitive to the calculation of the pore pressure by Eq. (7). Therefore, we estimate optimum values of \(c\) assuming values of \(r\) (i.e. depth), by computing correlation coefficients between observed velocity changes and modelled pore pressure changes. Since we expect decrease in seismic velocity due to increase in pore pressure, we determine the optimum value of \(c\) with the largest negative correlation. After optimum values of \(c\) are estimated at each station and depth, we construct a map of \(c\).