Storage in the deep aquifer: Stochastic process in time and frequency domain using groundwater level response to seismic activation of the Tohoku earthquake, Japan, 2011

This study delves into the interplay of Rayleigh wave‐groundwater‐aquifer dynamics, utilizing seismic data from designated stations, particularly the magnitude 9 event on March 11, 2011 (02:46:23 PM epicenter), Tohoku earthquake, Japan. Auto spectra uncover significant peaks, linking groundwater level with the vertical motion of Rayleigh waves. High coherence between groundwater levels and seismogram vertical displacements confirms this connection. Coherence analysis validates it at the significant frequency bands. Groundwater storage coefficient estimation aligns with typical values of confined aquifers, ranging from 1.0E‐05 to 1.0E‐03. This consistency applies across different methods and other earthquake events, highlighting seismic impact. The findings enhance understanding of aquifer behavior during seismic events, aiding groundwater management and earthquake readiness in susceptible regions. Further research should expand datasets for coefficient accuracy. Insights advance aquifer behavior comprehension during seismic events, guiding groundwater management and earthquake readiness in prone areas.


| INTRODUC TI ON
Groundwater, a valuable resource, is globally threatened by depletion.Assessing confined aquifer storage is vital for comprehending water movement.Groundwater levels respond to nearby surface water changes, especially near oceans or rivers.A time-frequency domain spectral approach evaluates aquifer hydraulic traits under tidal influences.Shih (2018) showed it computes hydraulic diffusivity using the transmissivity ratio to storage coefficient.The method employs water level data and boundaries to analyze larger-scale aquifers' hydraulic parameters.The techniques enhance groundwater insight and management strategies.Cooper Jr. et al. (1965) formulated models using groundwater levels and Rayleigh wave seismic displacement.Shih (2009) further highlighted the role of Rayleigh waves in the frequency domain, transmitting seismic waves to wells.Shih explored spectral density and coherence, exemplified by Eastern Taiwan's post-2004 Sumatra-Andaman Islands Earthquake study.Shih (2017) expanded this, analyzing ten earthquakes occurred along the East Asia-Western Pacific Ocean Ring of Fire, correlating groundwater storage with Quaternary aquifers.Rayleigh waves (7-25 s) from epicenters reached wells, revealing storage coefficients (1.0E-04 -1.0E-03).These studies underscore Rayleigh waves' groundwater dynamics insights and seismic evaluation of aquifer storage.
To detect earthquake seismic waves early, the Central Weather Bureau of Taiwan has strategically installed monitoring wells in representative regions, reaching deep aquifers.Simultaneously, the study also identify valuable information about groundwater storage by analyzing spectral coherence between groundwater and seismic vertical displacement.The approach presents an innovative and efficient alternative to traditional field testing methods for assessing groundwater resources.The research primarily aims to determine groundwater storage by correlating groundwater levels and seismic Rayleigh wave displacements monitored within groundwater wells.
The result provides insights into the region's subsurface hydrology, offering new understandings of intricate groundwater dynamics.By leveraging advanced monitoring techniques and exploiting the synergy between seismic and groundwater data, this study introduces a pioneering method to quantify groundwater storage.These results hold the potential to enhance knowledge of subsurface hydrological processes and guide improved groundwater management strategies in the study area.

| Background information
Taiwan lies on the western Pacific Rim within the Ryukyu-Taiwan-Philippines arc chain (Figure 1).Its tectonic evolution, influenced by crustal plate interactions, has led to distinctive Cenozoic sediments with a mobile or orogenic character.These sediments rest on a pretertiary metamorphic basement, overlain by tertiary sediments, forming a significant thickness (exceeding tens of thousands of meters) (Ho, 1986).The main island of Taiwan is at the convergence of the Eurasian plate from the west and the Philippine Sea plate, causing enduring seismic activity due to inherent tectonic traits.Taiwan's geological position within the Ryukyu-Taiwan-Philippines arc chain and its interaction with crustal plates have shaped its tectonic history.Mobile orogenic sediments and ongoing seismic activity underscore the region's dynamic nature.
The seismogram captures surface wave movement from the epicenter to the receiver, noting displacement in the east (E), north (N) and vertical (Z).This data shifts to radial (x), transverse (y), and vertical (w) displacement relative to the epicenter near the ground.
Interestingly, the Z-axis recorded groundwater level (h) aligns with Rayleigh wave's vertical displacement (w).Rayleigh waves exhibit out-of-phase motion in the x-w plane, forming a retrograde ellipse on the surface (Lay & Wallace, 1995;Stein & Wysession, 2009).This elliptical motion opposes the wave propagation direction at the ellipse's top.Similarly, the groundwater head in well-aquifer systems correlates with vertical land surface motion, as with Rayleigh waves (Cooper Jr et al., 1965;Shih, 2009).
This study collected six data sets, comprising groundwater head measurements and seismological records obtained from Taiwan's Central Weather Bureau (CWB) (Figure 1 and Table 1).The 2011 Tohoku earthquake, also known as the Great East Japan Earthquake, occurred on March 11, 2011, at 02:46:23 PM at the epicenter, off the northeastern coast of Honshu, Japan.It had a magnitude of 9 and was one of the most powerful earthquakes ever recorded.The earthquake triggered a massive tsunami that caused widespread devastation along the Japanese coast, significantly lossing of life and property.The efficacy of our proposed method relies upon the accessibility of potent Rayleigh waves serving as activation sources.
Given the considerable span of propagation around 2700 km, harnessing a seismic event of substantial magnitude is imperative.
While incorporating data from alternative seismic occurrences, encompassing those of moderate and minor scale, could indeed aid in evaluating the uncertainty of our study and affirm the dependability of our solutions, the central aim was to leverage a notably distinguished and universally acknowledged earthquake source.This choice served as a powerful means to showcase the latent capabilities of our methodology in unearthing groundwater storage within the uncharted depths of the aquifer.The groundwater wells selected for analysis included TUN and HWA in eastern Taiwan, as well as DON, LIU, NAB, and CHI, arranged sequentially from north to south in the western part of Taiwan.Initially, these wells were installed as part of earthquake alert research efforts in Taiwan.Geological logs indicated groundwater well depths within 130 to 222 meters, accessing confined aquifers (Table 2).Complementing groundwater monitoring, controlled seismogram stations matched monitoring wells: NANB for TUN, HWAH for HWA, WGK for DON, TPUB for LIU, SGS for NAB, and MASB for CHI (Table 1).These stations recorded seismic activity aligned with well-collected groundwater data (Figures 1 and 2).The method is rooted in large aquifer natural forcing, differing from conventional well-centered hydraulic testing, which assesses small ranges.

Significance statement
This groundbreaking study introduces a novel approach to estimating groundwater storage in the studied region by where is a time-dependent random variable; Γ is the complex Fourier components in the frequency domain; i = √ − 1; t is the elapsed time; and f is the frequency.Realistically, let the finite time be incorporated in Equation (1); it rewrites as where T is finite time sequence.
The transformed component Γ is not only finite in time length but is also frequency-dependent. Equation ( 2) is limited in its applicability within a small time range due to insufficient statistical significance.Fluctuated properties are disregarded when the duration of groundwater head and seismic signals is too short.
A stationary time series of a length of T, it divides into n d contiguous segments, with each segment length of T s , it estimates the two-sided auto spectra for each part by By averaging each resultant component, it obtains a final "smooth" two-sided auto spectra and should satisfy the statistical significance. (1) TA B L E 1 Information of groundwater and seismogram monitoring station.where the complex conjugate of Γ i f k is Γ * i f k , and k = 0, 1, … ., N ∕ 2. The smooth computation of cross-spectra and associated quantities are (Bendat & Piersol, 2011) where It gives a 95% confidence interval (CI) of the auto spectra where 2 n; is the Chi-square distribution for a percentage so that the probability  2 n >  2 n; =  with degrees of freedom n, and n = 2n d .The coherence with a 95% confidence interval is Shih et al. (1999) derived the 95% non-zero coherence significance level.

| Determination of aquifer storage
According to the report, the time-dependent expansion response representing harmonic components can be treated as a static problem in a confined aquifer characterized by moderate porosity and permeability.In this context, the influence of inertial effects can be disregarded (Bredehoeft, 1967).By considering the relationship between the Poisson solid of the groundwater matrix and groundwater storage, the study derived the spectral representation of the water level in the well and the vertical displacement of the Rayleigh wave at the specific location (Shih, 2009): where S is the storage in an aquifer; G ww is the auto-spectra of vertical displacement of Rayleigh waves; G hh is groundwater head; b is the aquifer thickness; and is the wavelength of Rayleigh waves.
In practical terms, the synthetic time series w(t) and h(t) represent the observed data driven by periodic components of earthquake activities and the water level in an aquifer, respectively.When analyzing these time series, if the components associated with Rayleigh waves can be identified, it signifies the presence of the relationship described in Equation ( 13).Auto spectra reveal meaningful signals related to the periodic time sequence influenced by the vertical displacement of Rayleigh waves or the water level in an aquifer (Shih, 2009).Additionally, cross-spectral density and coherence are valuable for assessing the strength of the relationship between these two processes in the time-frequency domain.A non-zero coherence level (NZC) indicates consistent measurements between the two processes in the frequency domain when cross-spectra are utilized.

| RE SULTS AND D ISCUSS I ON
The study employs Rayleigh wave and groundwater level records from designated stations (Table 1; Figure 2).The seismic epicenterto-groundwater well distances spanned 2400 to 2700 km.A prior study by Shih et al. (2013) examined essential groundwater head and seismogram characteristics were examined using descriptive statistics.The normality assessment employed the K-S test, indicating deviations from the normal distribution.It necessitated prior treatment for stationary spectral analysis.Auto spectra, with a frequency resolution of 0.781E-02 Hz and 95% confidence interval, ranged from 0.36 to 8.26.Cross spectra exhibited a non-zero coherence significance of 0.78.Given coherence, non-zero levels were used to identify significant peaks in cross-spectral density rather than confidence intervals.
Figure 3 displays the auto spectra for the seismogram's groundwater level and the radial (x) and vertical (w) components.Notably, all pairs of groundwater and controlled seismic displacement exhibit significant peaks, demonstrating periodic fluctuations within three dominant periods at various stations: 21.3, 25.6, and 32 s (see Table 3).Shearer (1999) previously reported that Rayleigh waves occur within the frequency range of approximately 20-300 s, with velocities ranging from 3 to 5 km/s.The estimated wave velocities in this study (Table 3) align with Shearer's findings, further validating the accuracy of the calculations.
(4) This study utilizes the auto spectra of groundwater head and vertical displacement, considering their wavelengths (Table 3) and the thickness of the aquifer (Table 2) to analyze aquifer storage using Equation ( 13).Through practical calculations, the storage coefficient is estimated to fall within the range of 1.0E-05 to 1.0E-03 (refer to Table 3).The range indicates the amount of water released from or taken into storage per unit head change, per unit surface area of the aquifer, at the target location.
It is worth noting that the estimated storage coefficient for the NAB site in this study (refer to Table 3) aligns with traditional values reported by Heath (1983) and Batu (1998) for confined aquifers, which typically range from 1.0E-05 to 1.0E-03.Additionally, the storage coefficient estimated for NAB in this study is comparable to the value inferred from the analysis of Earth tides, considering the barometric effect (Table 2).Different methods employed to estimate the storage coefficient for the NAB site consistently yield values on the same order of 1.0E-05.For the HWA site, the storage coefficient calculated based on seismic activity from the Sumatra-Andaman Islands earthquake on December 26, 2004 (Shih, 2009), and the present research on the Tohoku earthquake on March 11, 2011, are 6.184E-03 and 1.176E-03, respectively (Tables 2 and 3).These different earthquake activities yield storage coefficients on the same order of magnitude for the HWA site, emphasizing the consistency of the findings.
As Rayleigh waves and groundwater level record at selected stations, statistical analysis unveiled notable deviations, prompting prior treatment for stationary spectral analysis.With a 95% confidence interval, auto spectra exhibited significant peaks and analyzing the spectral representation of seismic Rayleigh waves and groundwater level.In addition, it highlights the comparison with other studies utilizing different methods, such as Earth tides and seismic activation by other earthquakes.The findings demonstrate the robustness of estimating aquifer storage through spectral representations and the response of groundwater head changes to vertical displacement caused by Rayleigh waves.This study significantly advances the understanding of groundwater dynamics, particularly concerning earthquake activity.Combining hydrogeological fluid dynamics with geophysical insights from seismology provides a unique perspective on subsurface processes.The implications of this research challenge conventional methodologies in hydrogeology and seismology, offering new insights and potential avenues for future investigations.With its interdisciplinary nature, this paper is sure to captivate the journal's readers, presenting compelling evidence and shedding light on the intricate interactions between hydrogeology and solid Earth processes.F I G U R E 1 Maps show the studied aquifers adjacent to the West Pacific Ocean.(Circle and triangle indicate groundwater and earthquake monitoring stations, respectively).[Colour figure can be viewed at wileyonlinelibrary.com]Spectral analysis Spectral analysis transforms periodic fluctuation time series into frequency domain power spectrum.Considering a time-dependent random variable in the time domain, (t) defines complex Fourier component as follows: as the sampling rate in time, the discrete frequency is where N denotes the length of segments.The smoothed, one-sided auto spectral density expresses as It computes the raw estimate cross-spectral density for each sub-record for two different time series, for example, (t) and (t) , through Fourier components Γ f k and Ψ f k utilizing Figure4illustrates the distinct high coherence between the groundwater level (h) and the seismogram's vertical displacement (w) within a specific frequency band.This coherence is supported by the corresponding periods listed in Table3.Notably, the MASB-CHI pair demonstrates the NZC at a dominant frequency; however, the coherence level is slightly lower than other station pairs.The study suggests that groundwater fluctuations are responsive to all stations and exhibit a genuine correlation with the vertical component of Rayleigh waves.
periodic fluctuations across the frequency band.The estimated wave velocities aligned with previous findings, validating the accuracy of the calculations.High coherence was observed between the groundwater level and vertical displacement of the seismogram, indicating a correlation with Rayleigh waves.The study estimated the storage coefficient of the aquifer, which fell within the range of 1.0E-05 to 1.0E-03.These values aligned with traditional values for confined aquifers and were consistent with estimates from Earth tides analysis.Various earthquake activities yielded similar storage coefficients for the target sites, highlighting the robustness of the findings.Overall, the study provided valuable insights into aquifer storage and its relationship with seismic activities, contributing to a better understanding of groundwater dynamics.The noteworthy findings for groundwater storage presented in this study are exclusive to this research and have not been observed elsewhere.TA B L E 3 Storage coefficient inferred from Rayleigh waves.Cross spectral density of groundwater level vs. vertical displacement due to Rayleigh waves.[Colour figure can be viewed at wileyonlinelibrary.com]

degree) Altitude b (m) Distance to the event (km) Geology of seismogram
a Friday, March 11, 2011, at 02:46:23 PM at the epicenter.b The altitude of groundwater is the level of top of the casing.TA B L E 2 Information of groundwater well and aquifer.Note: -, Lack of data; diameter of the well is 0.1524 m. a Shih (2009).b Shih (2018).