A Fourier series non-stationary coherency model with respect to site conditions for the horizontal component of ground motion

A Fourier series non-stationary coherency model with respect to the site conditions for the horizontal component was developed. The evolution of the model was achieved using parameters related to the site, inter-station distance, and time. First, based on the simulation of a non-stationary ground-motion field considering the wave-passage effect and site-response effect, the approach used for the estimation of coherency using wavelet transform is presented. Subsequently, the Fourier series non-stationary coherency model is formulated. The parameters are considered piecewise constant variables to represent the non-stationarity of the estimated coherency. The effects of the site and inter-station distance on the proposed model are presented. Parameters related to the site and inter-station distance were obtained. Finally, the proposed model is compared with the ground motion records at the SMART-1 array during Event 45, the stationary model, and the data from the Argostoli rock-site dense array. This indicates that the Fourier series non-stationary coherency model with respect to site conditions for the horizontal component can well match the correlation of the realistic spatially variable seismic ground motion, which is related to the site, time, and inter-station distance.


Introduction
The spatial variation of seismic ground motion is the amplitude and phase variation of the seismic ground motion of two stations within 100 m or more. The spatial variation of seismic ground motion significantly influences the dynamic response of long-span bridges (Saxena et al. 2000), nuclear power plants (Mohamed et al. 2015), oil and gas pipelines (Hindy and Novak 1980), transmission systems (Tian et al. 2018), and other long-span structures. The spatial variation in seismic ground motion is often characterized by spatial coherency, a complex-valued function that provides a quantitative measure of the correlation between two ground motions recorded at two adjacent stations.
Several studies have demonstrated that coherency is site-related. Chiu et al. (1995) found that the coherency estimated from SMART1 was similar to that from SMART2. Because of equal station distance, 5, 7, and 17 pairs from SMART2 were chosen for separation distances of 400, 800, and 1500 m. This data deficiency led to the conclusion that coherency between the SMART1 and SMART2 arrays indicated no significant differences. To overcome the lack of data from base rock dense arrays, Ding et al. (2004) proposed a coherency model for base rock sites based on the simulation of near-field ground motions. For convenience of calculation, some simplifications were made. This method might address the problem of insufficient data but requires further refinement, because it does not compare with existing records. Abrahamson (2006) compared the average coherency for soil sites, soft-rock sites, and hard-rock sites and found that coherency is related to the site. However, the proposed model did not contain components related to the site. Svay et al. (2017) compared the coherency estimated from the Argostoli database with Luco and Wong (1986) and the Abrahamson model (2006) (Abrahamson 2006) and found that these two models could not provide suitable answers. All the parameters were siterelated. Imtiaz et al. (2018b) compared the coherency estimated from the Argostoli rock-site dense array and the soil-site dense array. Imtiaz et al. (2018b) concluded that for frequencies below 5 Hz, coherency seems higher at the rock site than at the soft-soil site. The observed rock coherency was slightly lower than that of soil above 5 Hz. This could be attributed to soil heterogeneities associated with the rock site. Therefore, the current arrays provide only the data used to distinguish the differences. Each array has only one V S30 , which is not sufficient to obtain the relationship between the site and coherency. Establishing more arrays is expensive. The present study attempts to contribute to this area of research by Monte Carlo simulation (Deodatis 1996) for analyzing the coherence between two ground motions recorded at two adjacent stations. It aims to generate non-stationary seismic ground motions based on site, magnitude, and hypocentral distance, to estimate time-frequency coherence by wavelet transform, and to investigate the effects of magnitude, hypocentral distance, site, and inter-station distance on coherency.
In this study, a Fourier series non-stationary coherency model with respect to site conditions is proposed. This study begins with the simulation of a non-stationary ground-motion field considering wave-passage and site-response effects. The wavelet transform was used to estimate coherency using the generalized Morlet wavelets. Subsequently, the expression of the non-stationary coherency model is presented. The non-stationary model is achieved using parameters related to time. For convenience, the parameters were considered as piecewise constant variables. The effects of site and inter-station distance are presented. Parameters related to the site and interstation distance were obtained. The proposed model was compared with the ground motion records at the SMART-1 array during Event 45, the Hao et al. (1989) model, and the data from the Argostoli rock-site dense array.

Theoretical background
Coherency describes the cross-correlation between high-frequency strong ground motions within 100 m or more generated from a large-magnitude earthquake. Strong ground motions were assumed to be stationary in time. Therefore, the cross spectrum and autospectrum are obtained using the Fourier transform. The coherency is obtained as follows: where γ is the coherency, S jk (ω) is the smoothed cross-spectrum, S jj (ω) is the auto-spectrum at station j, and S kk (ω) is the auto-spectrum at station k. According to the definition of power spectrum density and cross-power spectrum, Priestley (1987) proposed the evolutionary power spectrum density and crossevolutionary power spectrum to describe the crosscorrelation and auto-correlation of non-stationary processes. Thus, the evolutionary coherency is obtained as where γ is the evolutionary coherency, S jk (ω) is the smoothed cross-evolutionary power spectrum, S jj (ω) is the auto-evolutionary power spectrum at station j, and S kk (ω) is the auto-evolutionary power spectrum at station k. A non-stationary stochastic process can be represented as follows (Priestley 1987): where A (ω, t) is the intensity and frequency modulating function, and dZ (ω) is the zero-mean, mutually independent, orthogonal increment process. The evolutionary power spectral density of f (t) is defined as The two non-stationary stochastic processes are represented as follows: The cross-evolutionary power spectral density is defined as where * denotes a complex conjugate.

Data and methodology
A Monte Carlo simulation (Deodatis 1996) was used to provide sufficient data for analyzing coherency. The non-stationary ground-motion field considers the wave-passage and site-response effects. After comparison, a total of 3000 Monte Carlo simulations were suitable. Stochastic variance tends to be stable, filling the range of variation and providing a stable database. More simulations bring too much computation. The time-frequency coherence can be estimated using the wavelet transform. The non-stationary coherency model can be fitted using the Fourier series to investigate the effects of the site and inter-station distance.
3.1 Simulation of non-stationary ground-motion field considering the wave-passage and site-response effects It is well known that spatially varying ground motions affect long-span structures. Previous studies have based their coherency models on seismic ground motions from a database of dense arrays. This method is particularly useful in studying the coherence between high-frequency strong ground motions within a few hundred meters generated from a largemagnitude earthquake with no strong motion dense array. The temporal non-stationarity of the amplitude of seismic ground motion can be obtained by multiplying a stationary process with the time-history intensity envelope function (Hao et al. 1989). It can only describe the non-stationary amplitude. To describe the non-stationarity of the amplitude and frequency of seismic ground motion, a time-frequency joint spectrum was proposed (Deodatis and Shinozuka 1988;Yeh and Wen 1990;Conte and Peng 1997;Wang et al. 2002;Rezaeian and Der Kiureghian 2008). However, this complex model requires many parameters to be determined, which hinders its engineering application.
A non-stationary model with temporal nonstationarity by the time-modulating function and spectral non-stationarity by the instantaneous frequency was proposed by Yu et al. (2015). The model can be combined with power spectral density (PSD) to generate an artificial seismic wave. The first PSD was proposed by Kanai (1957). The simulation of seismic ground acceleration was treated as filtered Gaussian white noise with a mean of zero. The most widely used PSD is the Clough-Penzien model (1975) (Clough and Penzien 1975). Priestley proposed an evolutionary power spectrum density for non-stationary seismic ground motion. Wang et al. (2019) estimated the evolutionary cross-spectrum and evolutionary coherency and presented a directional coherency model (Wang et al. 2019). These parameters were obtained from the ground motion records of the SMART-1 array. Yu et al. (2015) proposed a time-frequency modulating function. This function is determined by site, magnitude, and hypocentral distance. The stochastic model (Nigam 1965) and evolutionary power spectrum density (Priestley 1987) are combined with a time-frequency modulating function, and an artificial seismic wave at the first station X is as follows: An artificial seismic wave at the other station Y is shown here.
where B(t, ω) is the deterministic time-frequency modulating function, reflecting the time-frequency amplitude of the time history; more details can be found in Yu et al. (2015), ω is the circular frequency, Δω is an increment of the circular frequency, and φ k is a random phase that spreads from −2π to 2π uniformly. τ is the lag time for the arrival of the seismic wave at station X relative to its arrival at station Y, S is the power spectrum density, and the Clough-Penzien model (1975) (Clough and Penzien 1975) is used (Deodatis 1996). Yu et al. (2015) provided parameters for three soil conditions: site I (260 m/s < V s30 < 510 m/s), site II (150 m/s < V s30 < 260 m/s), and site III (V s30 < 150 m/s). The parameters of the Clough-Penzien model could be identified by three soil conditions (rock or stiff soil, deep cohesionless soil, and soft to medium clays and sands). These two are similar to sites C, D, and E; therefore, sites C, D, and E are used for the coherency analysis. Yaghmaei-Sabegh et al. (2014 and concluded that site classification based on only V s30 might be doubtful (Yaghmaei-Sabegh and Tsang 2014; Yaghmaei-Sabegh and Rupakhety 2020). A hopeful method that takes into account V s30 and horizontal to vertical spectral ratio curves was proposed (Yaghmaei-Sabegh and Rupakhety 2020). However, the site classification based on V s30 is widely used and easy to compare with other research results. It is used in this paper. According to Hao et al. (1989): where d XY is the inter-station distance between stations X and Y and v app is the apparent wave velocity. As shown in Fig. 1, in the database, the magnitude ranges from 4 to 8.5, the hypocentral distance ranges from 30 to 100km, the site ranges from C to D, and the inter-station distance ranges from 50 to 200.

The time-frequency coherency by the continuous wavelet transform
Because the previous coherency is based on the assumption that the seismic ground motion is stationary in time, the frequency content of the seismic motions can be precisely captured by ordinary Fourier analysis. However, the associated transform only provides the average spectral composition of the signal. Therefore, the short-time Fourier transform (STFT) was used (Cohen 1995). The entire time-domain process is decomposed into numerous small processes of equal length. Each small process was approximately stationary. Subsequently, the Fourier transform is used to determine the time-dependent frequency content. The STFT has certain limitations. For example, the narrow window function has good time resolution but poor frequency resolution, and vice versa. The inherent limitations of STFT can be overcome by wavelet transform (Grinsted et al. 2004). It has a lower time resolution and higher frequency resolution at low frequencies and a higher time resolution and lower frequency resolution at high frequencies.
Qiao et al. (2020)  Morlet (Grinsted et al. 2004) provided a balance between localization in time and frequency and was used in this paper. It is defined as where ω 0 is the dimensionless frequency, it is 0.6 for the Morlet wavelet. η is the dimensionless time. Compared with the Fourier transform, a Morlet is used to replace e −iωt . The wavelet transform allows researchers to capture the co-variation between two time series in both time and frequency domains. The wavelet function can be stretched by varying the wavelet scale(s) and translated by changing the localized time. Grinsted et al. (2004) proposed that the continuous wavelet transform of a time series(x n , n = 1, ..., N) with time steps δt is defined as the convolution of with the scaled and normalized wavelet: where δt is the time step, s is the wavelet scale, N is the length of a time series (x n , n = 1, ..., N), and ψ 0 is the Morlet. Torrence and Webster (1999) defined coherence as the square of the cross-spectrum normalized by the individual power spectra (Torrence and Webster 1999). Based on the above, Grinsted et al. (2004) introduced the definition of wavelet coherence and smoothing of the cross-spectrum (time and scale) (Grinsted et al. 2004). Smoothing of the crossspectrum should be performed to avoid a situation where the coherence is equal to one at all times and scales. Wavelet spectra: where W X n is the continuous wavelet transform of a time series at station X, W Y n is the continuous wavelet transform of a time series at station Y, and the * denotes the complex conjugation. Smoothing of the cross-spectrum should be performed to avoid a situation where the coherence is equal to one at all times and scales. Wavelet coherence: where W XY n , W XX n , and W Y Y n denote the wavelet spectra, S is the smoothing operator, and s is the scale. The entire artificial seismic wave was used to compute the coherency in this study (Ding et al. 2015). This study only estimated the coherence of the horizontal component of ground motion.

The Fourier series non-stationary coherency model
The simulated ground motion in this study was affected by magnitude, source distance, and site. Therefore, the lengths of the ground motions were also different. If the coordinate axis is time, the coherence lengths are different and cannot be compared with each other. Therefore, the normalized processing of the length of the ground motion was conducted. Therefore, normalized durations (0.1, 0.2, ..., 0.9, and 1.0) were used in this study. The entire nonstationary strong ground motion is divided into ten windows according to the moment when the cumulative energy is equal to 0.1, 0.2... 0.9, and 1.0. In each part, the parameters were estimated using the mean value of the corresponding calculated values at all moments of strong ground motions. Figure 2 shows the non-stationary coherency. At a certain normalized duration, the coherency decreases with an increase in frequency. For frequencies below 1 Hz, it is evident that coherency decreases with frequency at all moments. However, the coherency fluctuates up and down after 1 Hz. The variation in coherency differed at each moment. In any case, coherency is a function of frequency and time.
Existing coherency models are only related to two parameters: frequency and inter-station distance.
Other factors, such as the site, were not included in these models. Any periodic function can be expressed as an infinite series of sine and cosine functions. Fourier series have been widely used to fit complex functions; for example, the integral relationship between the straightness error and angle error (Ekinci and Mayer 2007), the surface curve of the guide rail scanned by an electron microscope (Tang et al. 2017), the relationship between single geometric error and displacement in a certain stroke of the moving axis of the machine tool (Niu et al. 2021), and the structure displacement function (Bao and Wang 2017a and b). After decomposition of the coherency by Fourier series, some parameters related to the site might appear. It was necessary to fit the extreme values of these points in the coherency model. A Fourier Fig. 2 The non-stationary coherency series non-stationary coherency model (FSNSCM) is proposed in this paper. γ (a 0 , a n , b n , k, ω) = a 0 2 + k n=1 [a n cos (nω) + b n sin (nω)] where a 0 , a n , and b n are the coefficients of the function and ω is the frequency. k is the order of the Fourier series. If k is too small, then the result is inaccurate. If k is too large, more parameters are obtained. After the comparison, k=4. Imtiaz et al. (2018a) found that magnitude (mostly below 5) might not affect coherency (Imtiaz et al. 2018a). Imtiaz et al. (2018a) explained that the magnitude examined in this study was mostly below five, so these earthquakes could be approximated by point sources (Imtiaz et al. 2018a). It can be concluded that when the inter-station distance between two stations is small compared to the hypocentral distance, the magnitude has no effect on the two ground motions at the two stations. Therefore, the effects of hypocentral distance and magnitude were not discussed here.

Results and discussion
4.1 Influence of the site on coherency Figure 3 shows a 3 varies with the normalized duration. The black line represents the variation in parameter for site C. The red line represents the variation in parameter for site D. The blue line represents the variation of the parameter for site E. To compare the effect of the site on a 3 , the mean value (Am) and variance (st) are calculated for each site. Therefore, the variance ratio a 3vr of a 3 under the three conditions can be calculated as follows: Sites C, D, and E were considered for the examination of site dependence for the same hypocentral distance (60 km), inter-station distances (100m), and magnitude (6) (Yu et al. 2015). In this study, the site affected the average shear-wave velocity (Lü and Zhao 2007), modulating function (Yu et al. 2015), and stationary power spectral density function (Clough and Penzien 1975). Based on Eqs. 1 to 10, the coherence was estimated, and the variance ratio of nine coefficients is shown in Fig. 4. In Fig. 4, the black square represents the variance ratio of the parameter for site C. The red circle represents the variance ratio of the parameters for site D. The blue triangle represents the variance ratio of the parameter for site E. The probability of rejection with significant difference was taken as 5%, and the coefficient of variation was about 1.960. This value was used to represent the boundaries. A coefficient was considered to be significantly affected by a condition if the variation of a coefficient under different conditions was outside this interval. Under the three conditions, the variance ratios of a 0 , a 1 , a 2 , a 3 , and a 4 were nearly the same and inside the interval. Therefore, the site may not affect these parameters. However, the variance ratios of b 1 were 4 and −4 for sites D and E, and nearly 0 for site C. The variance ratio of b 2 was approximately 3% for sites C and D and nearly 4 for site E. The variance ratio of b 3 was nearly 2 for sites C and D, approximately 16 for site E. The variance ratio of b 4 was approximately 4 for sites C and D, and approximately −1 for site E. If the variance ratio of each parameter is less than 1.960, the site might not affect each parameter. It is clear that the site might have an effect on these 4 parameters. These findings indicate that coherency is related to site. The coefficients for the three sites are listed in Tables 1, 2, and 3. a 0 a 1 , a 2 , a 3 , and a 4 are not related to the site, but are constants. b 1 , b 2 , b 3 , and b 4 are related to the site. Previous coherency models are decided by frequency and inter-station distance, so the coefficient in FSNSCM that is related to inter-station distance should be analyzed.

Influence of the inter-station distance on coherency
Most long-span structures range from 50 to 200 m. It was necessary to analyze the changes in each coefficient from 50 to 200 m. The curve of a 0 versus the inter-station distance is shown in Fig. 5(a). To compare the effect of inter-station distance on each parameter, the variance ratio was obtained basing on Eq. 18. The probability of rejection with significant difference was taken as 5%, and the coefficient of variation was about 1.960. This value was used to represent the boundaries. A coefficient was considered to be significantly affected by a condition if the variation of a coefficient under different conditions was outside this interval. Figure 5(b) shows that the inter-station distance has a greater effect on the variation in b 1 , b 2 , b 3 , and b 4 . Thus, b 1 , b 2 , b 3 , and b 4 are functions of inter-station distance. Polynomial fitting was performed using Eqs. 19,20,21,and 22; the coefficients are listed in Tables 4, 5, and 6.
where T is the normalized duration, d is the interstation distance, and p 11 , p 21 , p 31 , p 12 , p 22 , p 32 , p 13 , p 23 , p 33 , p 14 , p 24 , and p 34 are coefficients. From Table 7, it can be observed that b 1 , b 2 , b 3 , and b 4 are related to the site and inter-station distance. FSNSCM contains components related to the site.      Figure 6 presents the variation in coherence with the inter-station distance, normalized duration, and site. Figure 6(a) shows that the coherence decays with frequency and inter-station distance for the normalized duration of 0.2 and site C, because the two ground motions become uncorrelated at high frequencies and long inter-station distances (Zerva 2009). Figure 6(b) shows the coherence under three normalized durations (0.1, 0.4, and 0.7) for an inter-station distance of 150m and site C. The entire ground motion was used to estimate coherence. Traditional coherency models are developed based on strong motion windows of ground motions that are treated as stationary. However, ground motion has many frequency components (pressure waves, shear waves, and surface waves). The difference in the travel speed of these frequency components leads to a difference in the time taken to reach the same local site (Wang et al. 2019). This can cause non-stationary coherency between the two ground motions. The effect of the non-stationarity of ground motion cannot be ignored (Yeh and Wen 1990). Figure 6(c) shows the coherence under three sites (C, D, and E) for an inter-station distance of 120m and normalized duration of 0.7. The estimated coherence is in the sequence of first site C, next site D, and last site E at low frequencies. The decay with frequency of coherence estimated at site C is flatter and slower than the decay of the coherences estimated at sites D and E. Zerva (2009) concluded that the soft soil layer acts as a filter to pass over the low-frequency components of the incident bedrock excitation. The coherence of these sites decreases more quickly with frequency than the coherence of the motion recorded at the rock sites, which exhibits a smooth trend with frequency.

Comparison with the data from SMART-1 array and Hao model
In this study, the east-west (EW) and north-south (NS) horizontal accelerations recorded during Event 45 earthquakes at the SMART-1 station were selected, as shown in Table 8. The central station (C00) and inner circle stations (except for I01 and I07) were selected. Wang et al. (2019) : where a EW and a NS represent the east-west (EW) and north-south (NS) horizontal accelerations. Thus, the coherence can be estimated. Equation 16 and the coefficients in Tables 3 and 6 are used to estimate the coherence at site D. Figure 7 presents a comparison between the average coherence and the simulation value of coherence with the normalized duration being 0.1, 0.3, 0.7, and 0.9.
The model matched well with the coherency of the seismic records. FSNSCM performs better than the Hao et al. (1989) Figure 8 shows the comparison between coherence from Svay et al. (2017) and the proposed model with an inter-station distance of (a) 55m and (b) 100m. The coherence from Svay et al. (2017) was lower than that of the proposed model with an inter-station distance of 55m in Fig. 8(a). The coherence from Svay et al. (2017) was higher than that of the proposed model with an inter-station distance of 100m in Fig. 8(b). One reason is that the V S30 used in FSNSCM is 510 m/s. This is lower than that reported by Svay et al. (2017). Imtiaz et al. (2018a) concluded that when frequencies are less than 5 Hz, coherence seems higher on the rock site than on the soft-soil site. This could be attributed to soil heterogeneities associated with the rock site. Another reason is the coherence computed from the entire record in the FSNSCM. Svay et al.
(2017) computed the coherence using a strong motion Fig. 6 The variation of FSNSCM with a inter-station distance, b normalized duration, and c site window. Ding et al. (2015) concluded that a strong motion window resulted in higher coherencies; however, the overall trend and amplitudes were nearly the same as the coherencies from the entire record. Therefore, the coherence from Svay et al. (2017) should be higher than that of the proposed model with an inter-station distance of 55m in Fig. 8(a). Figure 9 shows the coherence with inter-station distance of 10m, 30m, 55m, and 100m from Svay et al. (2017). It is obvious that coherence decreased with an increase in inter-station distance. However, the coherence is nearly similar for inter-station distance of 55m and 100m. This might be due to data deficiencies that led to smaller than normal values. The results show the behavior of the coherence estimated from the simulation of non-stationary ground motion. The objective of this study was to propose coherency related to the site, inter-station distance, frequency, and normalized duration. The original data have the characteristics of up and down fluctuations. It is difficult to explain this result, but it might be related to non-stationarity. The 4th-order Fourier series chosen to model fit the data for the horizontal component well. It is constrained to have a value of unity at 0 frequency and decay up and down. It is defined by the following nine parameters. Five of these were determined by the site. Three of these were determined by the inter-station distance.

Conclusions
The Fourier series non-stationary coherency model with respect to site conditions describes the crosscorrelation of non-stationary strong ground motions generated from a large-magnitude earthquake within a few hundred meters. The model is based on the simulation of a non-stationary ground-motion field considering the wave-passage effect, site-response effect, and wavelet transform. All parameters are time-related to represent the non-stationarity of strong ground motions. The parameter values at each moment were calculated independently. For convenience, the parameters were considered piecewise constant variables. The model was compared with the coherence estimated by data from the SMART-1 array, Hao et al. (1989) model, and data from the Argostoli rock-site dense array. The proposed model follows the actual mean coherency of non-stationary strong ground motion records. This study identified the relationships between coherence as a function of site, inter-station distance, frequency, and time.
The present study lays the groundwork for future research on the effect of non-uniform seismic excitation on large structures such as bridges, pipelines, and transmission tower lines with extended foundations. The findings from this study can be used as an example of different places with no seismic ground motions from the database of the dense array. Further work is required to reduce the number of model parameters while ensuring the accuracy.