Preseismic, coseismic and postseismic water-level changes caused by the Gorkha earthquake are clearly identified in the seismic wave and water-level data (Figs. 3 and 4). Compared with the long-term water-level oscillation, the water-level show obvious coseismic impulses with short-term sustained postseismic variation mentioned as the duration times (see Fig. 1). The approximate durations of the short-term coseismic water level sustainable changes (the recovery times) are also measured by when the water level decreases down to 5% of the maximum coseismic response with the tidal response removal. The amplitudes of the coseismic water-level changes are also shown in Table 1 and Fig. 1. We calculate the difference between the levels within 2 h after the origin time of the Gorkha earthquake to get the amplitude of the oscillation in each well (Ma and Huang, 2017).

On comparing the seismic wave (Fig. 3) with the water level (Fig. 4), it is easy to distinguish the coseismic water-level response to the Gorkha earthquake in the D22 well and LJ well whose maximum amplitudes of 0.76 m and 0.16 m, respectively. The duration times of the coseismic water-level changes are 109 min and 97 min. By exact checking of the seismic-wave and water-level time windows, we can discover that the coseismic water-level response mainly corresponds to the Rayleigh surface-wave arrival times (see Figs. 3 and 4). These results suggest that the oscillation amplitudes and afterward durations of the coseismic water-level changes are associated with teleseismic surface-wave oscillations.

### Coseismic coherency analysis

To estimate how the seismic wave influences the water level, we first calculated the cross ordinary coherence functions \({\gamma }_{xy}^{2}\) among the water level, vertical velocity seismograms and barometric pressure for each station (Lai et al. 2013). The ordinary coherence function is defined as

$${\gamma }_{xy}^{2}=\frac{{\left|{G}_{xy}(\omega )\right|}^{2}}{{G}_{xx}\left(\omega \right){G}_{yy}(\omega )},$$

(1)

where \({G}_{xx}(\omega )\) and \({G}_{yy}(\omega )\) are the power spectra of two signals, respectively, and \({G}_{xy}(\omega )\) is the cross-power spectrum between them.

In the calculation, we use the data of each 3 days around the earthquake from April 23, 2015 to April 28, 2015. These data are continuous and stable. The time window and overlap are 1024 min and 512 min, respectively. We calculate the coherence before, around and after the earthquake origin time (Figs. 5 and 6). Previous studies show that the barometric pressure and Earth tide are two persistent factors affecting the water-level change (e.g., Lai et al. 2013; Zhang et al. 2019a, b). Because barometric pressure is one of the factors of well water-level change, the coseismic influence of seismic wave on water level in a well may be contaminated by it. We must exclude the possibility that barometric pressure has influence on the seismic wave, and determine the potential coseismic relationship between seismic wave and water level. The great transfer efficiency of semidiurnal tide for water level locates around 1–8 cpd. In the low-frequency band (*f* < 1 cpd), the barometric pressure also shows the great transfer efficiency for most wells whose ordinary coherence functions are up to 0.9. In the high-frequency band (*f* > 8 cpd), both efficiencies for water level decrease though (Lai et al. 2013). However, in the D22 well, we can observe the high efficiencies from barometric pressure, which may be caused by the in-well barometer. The coseismic shaking breaks this high coherence in this high-frequency band (Fig. 5c second panel), while the water-level response to velocity show the coherence enhancement between the preseismic and postseismic stages (see Figs. 5a, second panel). At the postseismic stage, the low-frequency (*f* < 8 cpd) transfer efficiencies of velocity from water level and barometric pressure increases (up to 0.7) in semidiurnal tide band which may represent in-phase responses of the semidiurnal tides and post seismic hydrological recovery. The coseismic coherence of barometric pressure and vertical velocity shows little downward at high frequencies. In the low-frequency band, the increased coherence may represent the tide transfer efficiencies in the both time series. In the LJ well, we can observe the similar water level in response to the velocity excited by the seismic waves at different stages (Fig. 6a). The barometric pressure transfer efficiencies to water level do not show obvious coseismic change as mainly affected in the 1–8 cpd (Fig. 6c), as previous mentioned (Lai et al. 2013). It is noted that the coherence between the velocity and barometric pressure show the similar coseismic decrease in the high-frequency band (*f* > 8 cpd) as that between the water level and barometric pressure in the D22 well (comparing Figs. 5c and 6b). An open well barometric pressure and seismic wave observation may make this correspondence. The relative high coherence except the coseismic stage may come from the wind noise. In short, the Gorkha earthquake coseismic responses directly may shake in high-frequency band (*f* > 8 cpd), and further switched the low-frequency semidiurnal tide transfer efficiencies (1–8 cpd), which further illustrate far field the hydrological responses to the great earthquake.

### Aquifer tidal response analysis

Among the tidal constituents, the O1 and M2 tides have large amplitudes and with low barometric effect. Therefore, they are the constituents most widely used in tidal analysis (e.g., Zhang et al. 2016; Wang et al. 2019). The M2 tide is more popular for its higher accuracy at 12.42 h period (Hsieh et al. 1987; Rojstaczer and Agnew 1989; Doan et al. 2006), which we apply for the following tide analysis.

The amplitude and phase responses of water level to the M2 tide can reflect aquifer storativity and permeability around well (Hsieh et al. 1987; Elkhoury et al. 2006; Doan et al. 2006; Xue et al. 2013). In a confined system, high permeability usually causes small phase lags, whereas low permeability results in large phase lags. The amplitude response is primary to measure specific storage (Zhang et al. 2016).

According to Cooper et al. (1965), the steady fluctuation of water level in a well occurs at the same frequency as the harmonic pressure head disturbance in the aquifer which, however, leads to different amplitude and phase shift. Hsieh et al. (1987) described the pressure head disturbance and water-level response as

$${h}_{f}={h}_{0}\text{exp}\left(i\omega t\right),$$

(2)

$$x={x}_{0}\text{exp}(i\omega t),$$

(3)

where \({h}_{f}\) denotes the fluctuating pressure head in the aquifer.\({h}_{0}\) is the complex amplitude of the pressure head fluctuation.\(x\) is the water level from the static position. \({x}_{0}\) is the complex amplitude of the water-level fluctuation. *t* indicates the time. \(\omega =2\pi /\tau\) is the frequency of fluctuation, where \(\tau\) is the period of fluctuation. The ratio between the amplitude of the water-level and that of the pressure head defines the amplitude response *A* as

$$A=\left|{x}_{0}/{h}_{0}\right|.$$

(4)

The phase shift is defined as

$$\eta =\text{arg}({x}_{0}/{h}_{0}),$$

(5)

where arg() is the argument of a complex number.

Figures 7 and 8 show the M2 tidal wave amplitude and phase response of water level in the D22 and LJ well, respectively. The blue error bars indicate the root-mean-square error (RMSE) of the tidal analysis. The bottom panels in figures (a) and (b) are the magnifications of the corresponding period of the gray background in the top panels, respectively. The M2 tidal signal is decomposed using the Baytap-G software based on Bayesian statistics (Tamura et al. 1991). Steps and spikes caused by instrument malfunctions or maintenance work are removed before the analysis. The barometric correction is performed by local barometric data. We use the data from February 8 to August 7 in 2015 when there were no other large earthquakes. The 10-day time window is selected to determine the amplitude and phase responses, since Elkhoury et al. (2006) have proven that using 240-h or 10-day time window can differentiate the M2 and S2 tidal constituents sufficiently. We have applied the Baytap-G software with different time windows to two different synthetic datasets (see Additional file 1: Text S1–S2 in the support information). The phase-shift estimations have proven that a 10-day time window is sufficient to separate the M2 and S2 tides and detect the phase changes (see Additional file 1: Figures S1–S4). Previous study (Shi et al. 2013) and our test may validate that the Baytap-G software with this fine time window selection can applied appropriately for the M2 tide analysis. In our data processing, the time label for each time window is the last day of the 10 days for the causal signal of the coseismic hydrological response which the past influences the present (Menke and Menke 2016). We thought in this way the coseismic changes can be reflected causally in time. Otherwise, there may be a delay for the standardization of coseismic variations in this small time-step analysis. We find during the coseismic days, the responses show significant variations that break the background long-term seasonal trend. The transient amplitude ratio decrease and the transient phase-shift increase in both wells caused by the Gorkha earthquake in two wells can be clearly distinguished and identified, which are consistent to the time-dependent coherence analysis. Near zero phase before the earthquake becomes positive after the earthquake for the D22 well (from 0° to 10°). While the positive phase before the earthquake becomes larger after the earthquake for the LJ well (from 7° to 9°), and restores at the postseismic phase. When the day of the coseismic well water-level response appears into the end of the time window, the changes of the response show in the fitting for amplitude and phase shift. The larger variations of amplitude ratio, and phase shift appears when the successive days after the earthquake go into the time window. But, we also find when the coseismic day moves out the beginning of the time window, the tidal response damps as the ordinary time. A complete successive causal impact may induce the intensive tidal responses in both amplitude and phase shift in the wells.