Application of spectral analysis to the reservoir-triggered earthquakes in Three Gorges reservoir region, China

In the Three Gorges reservoir region, central China, seismic activity increased substantially after the reservoir impoundment in 2003 which continues till date. Previous studies show that these are reservoir-triggered earthquakes and various factors are responsible for the increase in seismic activity after the reservoir impoundment. However, these studies do not provide a comprehensive assessment of influence of reservoir water level variations on spatiotemporal distribution of earthquakes. In this study, we statistically analyze the influence of the water level variations on the increased seismic activity in the reservoir region for the period from May 2003 to April 2020, using the power spectrum and singular spectrum techniques. Our statistical analyses confirm the influence of long-term variations in the water level time series on the occurrence of earthquakes after the reservoir impoundment. The analysis also indicates a positive role of annual reservoir water level fluctuations in the total seismicity of the region. Depending on the cluster patterns and relationship with faults, the earthquakes of the Three Gorges reservoir region are divided into three seismic zones (A, B, and C). For zone C, both the power spectrum and singular spectrum analyses confirm the strong periodic influence of reservoir water level variations on the earthquakes. Increase in seismicity of zone B is only in the initial period but not in the later stages of water impoundment and our statistical analyses indicate that the seismicity of this zone is not directly related to the annual reservoir water level variations. This confirms the conjecture in the earlier studies that the seismicity of this zone is related to the collapse of coal mines present in the area in the initial stages of reservoir impoundment. For zone A, our statistical analyses do not show strong influence of the annual reservoir water level variations on the occurrence of earthquakes. We suggest that this is due to the contribution of various other factors along with reservoir impoundment in the occurrence of earthquakes in this zone, as also opined in some earlier studies.


Introduction
Earthquakes caused by the impoundment of hydroelectric reservoirs, termed as reservoirtriggered seismicity (McGarr and Simpson 1997), have been extensively studied and concerned since the 1950s (Carder 1945;Gupta 1972;McGarr et al. 2014;Chen et al. 2023). Reservoir-triggered seismicity occurs due to various factors, including the state of regional stress field, regional geology, rock and fracture permeability, reservoir water level fluctuations, and reservoir dimensions, etc. Among these factors, water level fluctuations play a very important role. In general, the effects of water on earthquakes can be classified into two types: (1) Weakening effect due to the chemical influence of water on the faults: This effect may decrease the adhesion of faults and thus promote the fault instability (Talwani 1985;Ellsworth 2013;Foulger et al. 2018;Atkinson et al. 2020); (2) Loading and pore pressure effects: In this case, reservoir impoundment may alter the stress field around the fault, thereby induce earthquakes (Bourne et al. 2014). In addition, stress-induced pore pressure will also promote fault slips by reducing the normal stress on the faults (Reoloffs 1988;Gupta et al. 1972;Rajendran and Harish 2000;Liu et al. 2011;Nascimento et al. 2004;Gahalaut and Hassoup 2012, Gahalaut 2016, Gahalaut et al. 2016Ellsworth 2013;Gupta et al. 1972;Mikhailov et al. 2017;Grigoli et al. 2017;Braun et al. 2018). The detailed quantitative impact of reservoir water level variations on earthquakes is a hot topic both from a practical and a scientific perspective (Telesca et al. 2012a(Telesca et al. , 2012bTelesca et al. 2015;Kumar et al. 2012). For this, different techniques have been used in studies related to reservoir-induced seismicity (Selim et al. 2002;Gahalaut and Hassoup 2012;Gahalaut et al. 2016;Telesca et al. 2012Telesca et al. , 2015Hosseini and Rajaei 2012). Gupta et al. (1972) qualitatively compared the reservoir water level time series with the seismic activity in the Koyna region, India, from 1963 to 1967 and observed a temporal correlation of seismic activity with the reservoir water level with a certain time lag. Gupta (2001) also studied the temporal relationship between earthquakes with M4 + in 1973, 1980 and 1993 and annual reservoir water level and found a two-month delay between seismicity and rapid increase in water level. Telesca et al. (2012c) applied singular spectrum technique to study the relationship between local seismic activity and reservoir water level variations at Enguri high dam, Japan. Results show that the earthquake process is related to the quasi-periodic change of water level, which is mainly annual accompanied by some high harmonics. Telesca et al. (2015) observed that the southwest swarm activity near Pertusillo Lake, Italy, from 2002 to 2011 has an annual periodicity and is significantly correlated with annual fluctuations in the lake level. Telesca et al. (2020) studied the relationship between temporal variation of the seismic activity and the water level time series in the Minchevilkou reservoir, Azerbaijan by using the comprehensive analysis of periodogram, singular spectrum analysis and empirical mode decomposition. Results show that the temporal occurrence of earthquakes is certainly correlated with the annual cycle of the reservoir water levels. All these studies show that in a case of reservoir-triggered seismicity, it is imperative to verify statistically the relation between water level and earthquake time series to analyze the influence of reservoir impoundment on earthquake occurrence in a region.
The Three Gorges Reservoir is one of the largest hydroelectric power stations in the world, which is located in Yichang city, Hubei province, China (Fig. 1). The dam is 181 m high, and the maximum water level is 175 m. The reservoir is geologically located on the stable Yangtze platform (Li et al. 2009). The Huangling anticline (HLA in Fig. 1) and Zigui syncline (ZGB in Fig. 1) are the main structures of the upper Yangtze fold belt. The HLA is cut by some NW-trending faults and bounded by NNE-to N-S-trending faults in the west and NNW-trending faults in the east (Fig. 1). The Zigui basin is located in the west of HLA and surrounded by NE-and nearly EW-trending faults. The basin is mainly composed of Triassic and Jurassic strata (Zhang et al. 2019). Before water impoundment, the reservoir area exhibited low levels of natural seismicity (Zhang et al. 2019). However, seismic activity in the same area has increased significantly since the reservoir impoundment in 2003. Before the construction of the dam, an analog seismic network consisting of 5 stations was installed in 1958. In 2001, a digital seismic network composed of 24 stations was established to monitor the earthquakes in the reservoir area. All these stations were equipped with short-period seismometers. In 2012, the network was reformed to be 22 stations and some stations were equipped with broad-band seismometers. The digital network has now been operated for about 20 years and provided good amount of seismic data for our study.
Previous studies have suggested that a significant correlation existed between earthquakes and the reservoir water filling in the Three Gorges reservoir area (Zhang et al. 2018;Yao et al. 2017;Zhang et al. 2018). Yao et al. (2017) pointed out that the b value (~ 1.0) for the seismicity after reservoir impoundment was the typical feature of the reservoir induced seismicity. Zhang et al. (2018) used epidemic-type aftershock sequence model to study the influence of the water impoundment on the earthquakes and suggested that the seismic activity in the rapid loading stage was stronger than that in the unloading stage. However, the above studies do not give an assessment for the influences of water level on the earthquakes from the viewpoint of statistical analysis. The purpose of this study is to use the most updated 20-year-long earthquake catalog from the seismic network to perform a detailed statistical analysis to investigate the influence of Three Gorges reservoir water impoundment on the earthquakes. The structure of the article is organized as follows. In Sect. 2, we provide the details related to water filling history and seismic activity in the reservoir area. Section 3 provides the theoretical basis for the analyses. The relationship between the seismic activity and the water filling history is examined by power spectrum

Seismicity and water level
For further analysis, the earthquakes from May 2003 to April 2020 (as shown in Fig. 1) are relocated using the double-difference method given by Waldhauser (2000). The velocity model is adopted from Li et al. (2009). The standard errors of the horizontal position and focal depth of the epicenters are less than 1 km, and 1.5 km, respectively. Figure 2a shows the spatial distribution of relocated earthquakes in the reservoir area. Most of the earthquakes are concentrated along the Yangtze river and mainly focused in Badong, Xietan and Zigui area (as defined in Fig. 1). To determine the completeness of the seismic data set, the magnitude of completeness (Mc) is estimated with the ZMAP software (Wiemer and Wyss 2000). As shown in Fig. 2b, Mc of all earthquakes in Fig. 2a is 1.0. Based on the clustering patterns and the correlation with the faults, the seismicity is divided into three zones (Fig. 2a), i.e., zone A (Badong, BD in short), zone B (Xietan, XT in short) and zone C (Zigui, XNS in short) as done in several earlier studies (Yi et al. 2012;Hua et al. 2013). Mc of the earthquakes in the three zones is 1.0, 0.7 and 0.8 ( Fig. 2c-e). Totally more than ten thousand earthquakes occurred in the three zones. About 93% of the earthquakes are micro-earthquakes with magnitude of 0.0 ~ 1.9. The number of earthquakes with M2.0 ~ 2.9, M3.0 ~ 3.9 and M4.0 + was 788, 71 and 9, respectively ( Table 1). The largest   (Fig. 3). After that, the frequency of the earthquakes declines rapidly until March 2016. As a whole in this period, the earthquake frequency increases followed by a decrease. Figure 4 shows the temporal variations of monthly earthquake frequency in the three zones during the three impoundment stages. In the stage I, the seismic activity in zone A increased instantaneously (Fig. 4a), and the earthquake frequency is significantly higher than that in the zones B and C ( Fig. 4d and g). During the stage II, there is a further increase in the cumulative number and frequency of the earthquakes in all the three zones (Fig. 4b, e and h). In the stage III, the frequency of seismic activity increased significantly in zones A and C in comparison with the previous two stages (Fig.4c and i), but the maximum monthly earthquake frequency in zone B is less than that during the stage II (Fig. 4f). No earthquakes with M ≥ 4.0 occurred in zone B during all the three stages. Also, no earthquakes with M ≥ 4.0 occurred in the stages I and II in the zones A and C. While during the stage III, 5 earthquakes of M ≥ 4.0 occurred in zone A and 4 earthquakes occurred in zone C.

Methodology
To verify the influence of water impoundment on the earthquakes in the Three Gorges reservoir region, we need to examine the periodicity in the water level time series and earthquake time series. For the purpose, power spectrum and singular spectrum analyses of the two time series are performed in this study.
The power spectrum is the power distribution of a time-domain signal over frequency, which is the Fourier transform of the auto-correlation sequence. The method can help us understand the periodic /frequency distribution of the water level and earthquake time series. Estimation techniques include parameters and nonparametric methods. Parametric methods are based on parametric models, such as auto-regressive (AR) spectral estimation, moving average (MA) spectral estimation, and auto-regressive moving average (ARMA) spectral estimation. Periodogram is one of the common nonparametric techniques (Akselrod et al. 1981), which takes the Fourier transform of the auto-correlation estimate and results in an estimate of the power spectrum.
Singular spectrum analysis is a statistical technique to process nonlinear time series, which is based on decomposition theory, and suitable for our study to learn about the periodic oscillation behavior of water level and earthquake frequency. It can decompose the original signal into the sum of independent and interpretable components, such as long-term trends, seasonal trend oscillating components and unstructured noise (Hassani 2007;Wu et al. 2020). Since the technique decomposes the entire sequence into different principal components, contributions from different periods can be estimated. The variance diagram of the time series shows the percentage contribution of each component in the original time series. In general, the variance of the initial components and the power spectrum contains information about the dominant period (Yadav et al. 2015;Gahalaut et al. 2016). The basic singular spectrum algorithm consists of two complementary stages, the decomposition of time series and the reconstruction of a desired additive component. Based on the observed time series, the trajectory matrix can be calculated and decomposed. After reconstruction, the signals representing different components can be extracted to show the long-term trend, periodicity and noise (Yuan et al. 2021). The three steps of algorithm are as follows.

Calculation of the trajectory matrix
The trajectory matrix X of the given time series is calculated according to the window length L, and the trajectory matrix X is in the order of L × K, where K = N-L + 1, and 1 < L < = N/2.
where the element x ij at (i,j) of the matrix X equals to x i+j−1 , that means all the elements on the inverse diagonal are the same. In addition, the window length L should not exceed 1/3 of the data length. (1)

Singular value decomposition and grouping.
Singular value decomposition is performed on the trajectory matrix, and the matrix S = XX T is defined as the transpose matrix of X. The eigenvalues of S are calculated and arranged in descending order 1 ≥ 2 ≥ ⋯ ≥ M ≥ 0 , the corresponding eigenvectors are U 1 , U 2 , ⋯ , U M . Assuming that d = rank(X), V k = X T U k ∕ √ k (k = 1, 2, ..., d) , U k and V k are eigenvectors of the trajectory matrix, U is the temporal empirical orthogonal function, and V is called the temporal principal component. The matrix X can be synthesized by the elementary matrix X i : is the singular spectrum. The maximum eigenvalue corresponds to the maximum eigenvector, which represents the maximum variation trend of the signal, while the eigenvector with the smaller eigenvalue is generally regarded as noise (Song et al. 2020).

Reconstruction
Reconstruction is to transform the primary matrix X obtained above into a new time series with a length of N, and the sum of all reconstructed components equals to the original data series. It consists of two stages: grouping and reconstruction. Grouping is to split the matrices into several groups by singular value decomposition and then summed up, while reconstruction is a diagonal averaging. Each set of data represents a certain characteristic of the original sequence, such as a long-term trend, a seasonal trend, or a noise signal. Z = X k is defined as the time series obtained by diagonal averaging of Z.

Data analysis and results
In the following sections, we perform detailed statistical analysis to study the influence of the reservoir water impoundment on the seismic activity in the Three Gorges reservoir area. (2)

Power spectrum analysis
As a preliminary analysis, power spectra of the water level time series (Fig. 5a) and the all_earthquakes time series (Fig. 5b) are calculated using periodogram technique. The purpose of the analysis is to obtain the frequency content of the two time series to check the effects of annual water level variations on the frequency of the earthquakes. The spectrum of the water level time series shows a dominant annual peak superimposed on a power law with spectral exponent b_water ~ 1.62, which is consistent with the features of annual variations in reservoir water level (Fig. 5a). The power spectrum of the all_earthquakes time series contains an annual cycle (Fig. 5b). The results of the power spectrum analysis for the earthquakes in the three zones for the entire time period, i.e., from May 2003 to April 2020 are shown in Figs. 5c-e. The power spectrum of the earthquake time series for zone C (Fig. 5e) shows a dominant annual variation. In zone A, annual cycle is also present (Fig. 5c), which is otherwise absent for the earthquake time series of zone B (Fig. 5d). The above power spectrum analysis does not indicate the existence of annual frequency in all the cases in Fig. 5. Due to the contribution of many different processes in the occurrence of earthquakes in the reservoir area, the time series may be composed of cycles other than annual cycles (Telesca et al. 2010;Yadav et al. 2015;Gahalaut et al. 2016). Therefore, to examine the contribution of other processes along with periodic variations of water level, other periodicities must be deducted. This can be done by separating the signal based on its pattern over time. Thus, in order to further quantify and to examine the impact of reservoir water level fluctuations on the earthquakes in detail, singular spectrum analysis is performed for the all_earthquakes and earthquake time series of the three zones along with water level time series.  Figures 6 and 7 show the results of the singular spectrum for the water level and the all_ earthquakes time series, respectively. From left to right, panels show the variance of components in percentage, reconstruction of decomposed components with higher variance, and their corresponding power spectra. The window length of the time series used in the analysis is 72 (360/5, 360 denotes the annual cycle, and bin size is 5 days). From the variance graph, it can be seen that except for the first four principal components other components share insignificant variance. This is the reason the first four components are considered in this analysis.

Singular spectrum analysis
For the water level time series the first reconstructed component, which shares 53.2% of the total variance, mainly contains long-period harmonics (Fig. 6c). Both the second (20.9% variance) and third (19.5% variance) components contain dominant 1-year cycle (Fig. 6f). For the first reconstructed component of the all_earthquakes time series, with a variance of 43.8%, has the long-period harmonics (Fig. 7c), while the second and the third components, with a variance of 5.0% and 3.3% (Fig. 7f), contain a dominant annual frequency (Fig. 7f). Figure 8 shows the singular spectrum analysis results for the earthquakes in the three seismic zones. Results for the seismic sequence in zone A show that the first component  (Fig. 8a) has an overall dominant long-period cycle along with a very weak annual cycle. Annual component is present in the second (7.5% variance) and third components (5.0% variance) along with a dominant long-period cycle (Fig. 8c). For the earthquake time series in zone B, the variance of the first component is 32.9% (Fig. 8d), and it has dominance of long-period cycles (Fig. 8f). The variance of the second, third and fourth components is 4.9%, 2.7%, 2.0% (Fig. 8d) and annual cycle is absent from all these three components (Fig. 8f). In the case of the earthquake time series of zone C, the first reconstructed component with a variance of 19.3% (Fig. 8g) shows an overall dominant long-period cycle (Fig. 8i). The second component with a variance of 5.3% has a dominant annual cycle, while the third component with a variance of 4.7% (Fig. 8g) has a dominant annual cycle along with some short period (less than 1 year) peaks.

Discussion
Seismicity has lasted for more than two decades in the Three Gorges reservoir area and a close relationship between water impoundment and earthquakes also has been defined. In the following sections, we will discuss the influences of water impoundment on the earthquakes from the viewpoint of statistical analyses.

Long-term effect of water level fluctuations
In the singular spectrum analysis, the power spectra of the first component for the water level and the all_earthquakes time series (Figs. 6c and 7c) show similar scaling behavior The singular spectrum analysis of the water level time series indicates that the first reconstructed component has a long-term period cycle (Fig. 6c). Similarly, the first reconstructed component of the all_earthquakes time series and of the earthquakes in the three zones consists of long-term periodicity (Figs. 7c and 8c). Before September 2008 during the initial reservoir filling (stage I and stage II), the seismicity is characterized by an increasing trend; while after stage II, during the periodically stationary high value of the water level time series (stage III), the first reconstructed component of the all_earthquakes and the earthquakes in zones A and C is characterized by the periodic changes (Fig. 9). In brief, the earthquakes in zones A and C show a similar pattern as in water level time series in the first reconstructed component which shares the maximum variance. This is not the case for zone B, where the seismic activity shows an overall decreasing trend after the stage II of the reservoir impoundment. It may be noted that the seismic activity in zones A and C accounts for 90% of the total seismic activity in the reservoir area. It is very clear from the singular spectrum analysis that the dominant long-term changes of the water level have strong influence on the seismic activity in zones A and C.

Annual effects of reservoir water level
In addition to the first component, the comparison between the other reconstructed components shows following features: (i) The second and third components of singular spectrum analysis display the presence of annual periodicity in the water level and the all_earthquakes time series (Fig. 6f and Fig. 7f), which indicates the influence of reservoir impoundment on the earthquake occurrence in the Three Gorges reservoir region; and (ii) singular spectrum analysis of the earthquakes in zone B shows the complete absence of annual frequency in the reconstructed components (Fig. 8f). Same is the case in the power spectrum analysis (Fig. 5d). In the previous studies, it was noted that the seismicity of this zone is mainly located in the Xietan coal mine area (Zhang et al. 2018(Zhang et al. , 2019. In the initial period of reservoir filling, especially during the period of stages I and II, collapse of mines might have induced seismic activity. Thus, the seismicity of zone B does not come under the categorization of reservoir triggered seismicity in the purview of physical mechanism and/or characteristics of reservoir triggered seismicity (Gahalaut 2021). This may be the reason for the absence of annual cycle in both the statistical analyses. (iii) Singular spectrum analysis and power spectrum analysis for seismic activity of zone A indicate not very clear annual periodicity (Figs.5c and 8c). In previous studies (Yao et al. 2017;Zhang et al. 2018), zone A seismicity has been associated with various factors like fault rupture, karst collapse, landslide and unloading of shallow surface (Yao et al. 2017;Zhang et al. 2018). Involvement of various factors in the occurrence of earthquakes may be the reason for weak annual periodicity in our statistical analyses. (iv) In some previous studies (Yao et al. 2017;Zhang et al. 2018Zhang et al. , 2019, zone C seismicity has been associated with the Fairy mount fault and the fault rupture instability due to the water level fluctuations is considered as the dominant factor in the increase in seismicity of this zone. Dominance of annual periodicity in the power spectra of second and third reconstructed components of singular spectrum analysis and also in the power spectrum analysis for zone C (Fig. 8i and 5e) clearly indicates the positive influence of reservoir impoundment on the seismicity of this zone. Fig. 9 A compilation of power spectra curves of the 1 st reconstructed components (from Figs. 6,7,8) for the time series of water level, all_earthquakes and the three zones. Two red dashed vertical line separates the three stages of water level same as in Fig. 3

Conclusions
Our detailed statistical analyses reveal the complicated relationship between water level variations and the earthquakes after reservoir impoundment in the Three Gorges reservoir. Effects of long-term variations in the reservoir water levels on the occurrence of earthquakes are very clear in the singular spectrum analysis. The presence of the annual cycle in the power spectrum and singular spectrum analyses for the all_earthquakes time series attests the overall positive role of the reservoir impoundment in the increase in seismicity after impoundment. Both the analyses suggest that the annual water level fluctuations certainly triggered seismic activity in the zone C of the reservoir area. In the zones A and B of the Three Gorges reservoir area, the earthquake genesis is more complex, and thus, the influence of annual fluctuations of water level on the earthquakes is not very apparent.