Simulation of Strong Earthquake Ground Motions Based on the Phase Derivative

Abstract

A physical method for modeling the phase spectrum of earthquake ground motion is derived by defining relationships between the envelope delay and Fourier amplitude. In this method, two parameters with clear physical meanings, namely the median arrival time and strong shock duration, are introduced. These parameters provide a logical basis for modeling the phase spectrum in a physical sense. A simulation method for earthquake ground motions is introduced, based on a physical amplitude model and the proposed method for modeling the phase spectrum. To investigate the physical meaning of the phase spectrum of earthquake ground motion and to be used for simulating earthquake ground motions, two techniques based on the discrete Fourier transform (DFT) and the continuous Fourier transform (CFT) are employed to calculate the envelope delay. It is demonstrated that when using the DFT, the range of envelope delays is dependent on the duration of the earthquake ground motion, and the range of envelope delays corresponding to peak amplitudes is dependent on the time span of the strong shock in ground motions. This dependency is not observed with the CFT. The proposed simulation method for earthquake ground motions was used to regenerate two recorded earthquake acceleration time histories. Numerical results demonstrate that this method can accurately reproduce the main characteristics of strong earthquake ground motion recordings.

1. Introduction

The modeling and synthesis of stochastic earthquake ground motions in the time domain are crucial for seismic response analysis and the reliability evaluation of engineering structures. Earthquake ground motion models can be classified into two categories based on whether they incorporate phase spectral characteristics. Models that do not consider phase spectral characteristics construct earthquake ground motions by filtering and windowing stationary processes and combining suitably scaled accelerograms [1,2,3,4,5,6]. A widely used model of this kind was proposed by Boore, which includes the source, propagation path, and site to model the amplitude spectrum [3,4]. A more recent model by Rezaeian and Der Kiureghian is a complete empirical model [6]. In these models, non-stationary earthquake ground motions are simulated using the spectral representation method with amplitude modulation. This method synthesizes the time histories of stationary ground motions using a trigonometric series with uniformly distributed phases and introduces temporal non-stationarities by applying a uniformly modulated function in the time domain [7,8,9].

It is recognized that previous research has focused extensively on the characteristics of the amplitude spectrum, with less attention given to the phase spectrum. However, the phase spectrum is crucial for capturing the non-stationary characteristics of earthquake ground motions [10,11]. The use of independently and uniformly distributed phase angles in the spectral representation method consistently produces stationary ground motions. The conventional treatment of earthquake ground motion phase angles has historically involved treating them as independently and uniformly distributed. This approach was motivated by the apparent uniformity observed in their statistical histograms. However, it is important to recognize that this perceived uniform distribution is a consequence of the inherent periodic nature of the phase angles.

In 1979, Ohsaki identified that the phase differences in earthquake ground motion followed a distribution closely resembling the normal distribution. Additionally, the shape of the seismic wave corresponded to the statistical histogram of phase differences observed within the same seismic wave [12]. Following this discovery, extensive efforts were undertaken to elucidate this intriguing relationship further and to devise methodologies for modeling the phase spectrum based on the distribution of phase differences [13,14,15,16,17]. Significantly, Jin and Liao clarified the physical significance of a phase difference function, often referred to as the relative arrival time of a wave group [15]. Furthermore, Thráinsson and Kiremidjian established a link between phase difference and Fourier amplitude, presenting a novel approach to modeling the phase spectrum [18,19]. Boore introduced the concept of envelope delay, quantified in units of time, which exhibits a pronounced resemblance to phase difference. Boore also explored the relationship between envelope delay, amplitude, and frequency [20]. Ding et al. conducted a systematic exploration of the physical properties inherent in the Fourier phase spectrum of earthquake ground motions, emphasizing the temporal characteristics of the phase spectrum [21]. The research studies mentioned above suggest that the phase difference of earthquake ground motion significantly influences the time-domain non-stationarity of these motions. Additionally, they uncover a relationship between the phase spectrum and amplitude spectrum.

Building upon the aforementioned findings regarding the phase spectrum, earthquake ground motions were generated utilizing a specific probability distribution of phase difference and an existing amplitude spectrum. Ohsaki [12] and Sato [10] synthesized ground motions by employing a phase difference spectrum in which each phase difference conforms to a normal distribution. Han et al. supposed that the phase derivative follows a normal distribution [22]. Yazdani et al., on the other hand, partition the frequency domain into segments, with the phase difference within each frequency band considered to follow a normal distribution [23]. Some researchers have alternatively suggested that a log-normal distribution provides a better fit for the phase difference [24,25]. Baglio proposed using the logistic distribution for the phase derivative, taking into account the presence of fat tails in phase derivatives [26,27]. Subsequently, several scholars have employed logically distributed phase differences to simulate ground motion [28,29]. However, it is noteworthy that these methods of earthquake ground motion simulation, while emphasizing the role of the phase spectrum in capturing the non-stationary characteristics of ground motion, often overlook the relationship between phase difference and amplitude—a relationship that has been previously identified in the literature [18,19,20]. Thráinsson and Kiremidjian simulated the phase differences conditionally based on the Fourier amplitudes, which were segmented into three distinct categories: small, intermediate, and large. Each amplitude category is associated with either a beta distribution or a combination of a beta distribution and a uniform distribution for defining the phase differences [18,19]. Though the relationship between phases and amplitudes is utilized, the categorization of amplitudes leads to an increase in the number of parameters, resulting in complexity and potentially hindering model interpretability and generalization.

This study aims to explore the significance of the phase spectrum in the non-stationary features of earthquake ground motion. Additionally, it seeks to introduce a new simulation approach for earthquake ground motions, leveraging the interplay between phase and amplitude spectra. In Section 2, the importance of the phase in earthquake ground motions is examined through a numerical experiment. Section 3 discusses the physical interpretation of the circular frequency-dependent phase derivative and introduces a calculation method for the circular frequency-dependent phase derivative based on the discrete Fourier transform. Section 4 analyzes the relationship between the phase derivative and duration to elucidate the temporal properties of the phase spectrum. In Section 5, a simulation method for earthquake ground motions that is grounded in the properties of phase and the relationship between phase derivatives and amplitudes is introduced. Discussions are presented in Section 6, and Section 7 presents the concluding remarks of this study.

2. Significance of Phase in Earthquake Ground Motions

To assess the significance of the phase spectrum, Sato conducted an insightful numerical experiment wherein the phase spectra of two recorded accelerograms from the Kobe and Kushiro earthquakes were interchanged [10]. Building on this foundation, a more detailed numerical experiment was carried out in this study to elucidate the mechanism of the phase spectrum using two accelerograms exhibiting significant differences in amplitude spectrum and waveform. The details of the two accelerograms are presented in

Table 1. Details of the two accelerograms used in the numerical experiment.

No.	Earthquake	Date	Station	Azimuth
1	Northridge-01	17 January 1994	Montebello–Bluff Rd.	206°
2	Parkfield-02, CA	28 September 2004	Parkfield–Cholame 1E	90°

, and their amplitude spectra are depicted in Figure 1.

The numerical experiment involves transforming the Northridge-01 and Parkfield-02, CA accelerograms into amplitude and phase spectra, respectively. Subsequently, two new acceleration time histories are synthesized. One is generated by applying the inverse Fourier transform, integrating the amplitude and phase spectra of the Northridge-01 and Parkfield-02, CA accelerograms, respectively. The other time history is generated similarly but integrates the amplitude and phase spectra of the Parkfield-02, CA and Northridge-01 accelerograms, respectively.

The two original accelerograms and the newly generated acceleration time histories are shown in Figure 2, Figure 3 and Figure 4. Specifically, Figure 2 and Figure 3 depict the newly generated acceleration time history integrating the amplitude spectrum of Northridge-01 and the phase spectrum of Parkfield-02, CA. Meanwhile, Figure 4 displays the other newly generated acceleration time history. As shown in Figure 2 and Figure 3, the waveform of the newly generated acceleration time history closely resembles that of the Parkfield-02, CA accelerogram, as their spectral components are identical due to the same amplitude spectrum as the Northridge-01 accelerogram. Figure 3 presents the newly generated acceleration time history with the Parkfield-02, CA accelerogram in a single-coordinate system. A strong agreement between the two time histories is evident, both in the overall waveform and in the prominent pulse. It is noteworthy that only the phase spectrum of the newly generated acceleration time history matched that of the Parkfield-02, CA accelerogram. This observation suggests that the Fourier phase spectrum plays a pivotal role in the time non-stationarity of earthquake ground motions.

Additionally, Figure 4 displays the other acceleration time history, which is composed of the amplitude and phase spectra of the Parkfield-02, CA and Northridge-01 accelerograms, respectively. Upon analysis of Figure 4, it can also be concluded that the phase spectrum is intricately linked to the non-stationary vibrations observed in earthquake ground motions. The results of the above numerical experiment underscore the significance of the phase spectrum in shaping the time history of ground motion, highlighting the essentiality of further research into phase characteristics.

3. Circular Frequency-Dependent Phase Derivative

The additive inverse of the circular frequency-dependent phase derivative, also known as envelope delay or relative arrival time of wave group, is represented by the following mapping formulation [30]:

$$t_e(f)=-\frac{\mathrm{d}\phi (\omega )}{\mathrm{d}\omega }=-\frac{1}{2\pi }\frac{\mathrm{d}\phi (f)}{\mathrm{d}f}$$

(1)

where $\phi (f)$ denotes the phase spectrum of earthquake ground motion. The value of $t_e$ is not unique due to the non-uniqueness of $\phi (f)$. When $\phi (f)$ changes in integer multiples of $2\pi$, it has no impact on the time series of earthquake ground motion. Boore proposed a method to calculate $t_e$ using a Fourier transform without first unwrapping the phase, thereby ensuring the uniqueness of $t_e$ [20]. This technique is referred to as the CFT method in this study, originating from the continuous Fourier transform. Another technique proposed in this study to ensure the uniqueness of $t_e$ is called the DFT method, originating from the discrete Fourier transform.

3.1. CFT Method

Given the time series of earthquake ground acceleration, denoted as $a(t)$, its Fourier transformation, $F(f)$, utilizing the continuous Fourier transform can be expressed as:

$$F(f)={\displaystyle {\int }_{-\infty }^{+\infty }a(t)e^{-\mathrm{i}2\pi ft}dt}=R(F)+\mathrm{i}I(F)=A(f)e^{\mathrm{i}\phi (f)}$$

(2)

where $f$ denotes the natural frequency; $R(F)$ and $I(F)$ are real and imaginary parts of $F(f)$, respectively; $A(f)$ and $\phi (f)$ denote the amplitude spectrum and phase spectrum of $a(t)$, respectively. Based on the CFT method, the envelope delay can be calculated by

$$t_e(f)=-\frac{1}{2\pi }\frac{\mathrm{d}\phi (f)}{\mathrm{d}f}=-\frac{1}{2\pi }\frac{\frac{\mathrm{d}I(F)}{\mathrm{d}f}R(F)-\frac{\mathrm{d}R(F)}{\mathrm{d}f}I(F)}{R^2(F)+I^2(F)}$$

(3)

It is observed that when $a(t)$ is provided, $R(F)$, $I(F)$, and $f$ remain constant, despite $\phi (f)$ altering under the condition that $a(t)$ remains unchanged. Consequently, a contradiction arises. The intermediary portion of Equation (3) will vary with shifts in $\phi (f)$, whereas the latter section remains unaffected. Nonetheless, the CFT method presents a means to determine the envelope delay of ground motions by effectively addressing these inconsistencies and providing a reliable approach for analysis.

3.2. DFT Method

Another method to calculate $t_e$ needs to determine the phase first. The recorded earthquake ground motions are not continuous but discrete. Therefore, the discrete Fourier transform is frequently employed for the spectral analysis of recorded earthquake ground motions. The discrete time series of earthquake ground acceleration, denoted as $a_n(t)$, can be expressed by the superposition of harmonic waves with different frequencies utilizing the discrete Fourier transform.

$$a_n(t)={\displaystyle \sum _{k=1}^{N}A\left(f_k\right)\cos \left(2\pi f_{k}t+\phi \left(f_k\right)\right)}$$

(4)

where $N$ represents the number of data points in $a_n(t)$; $A\left(f_k\right)$ and $\phi \left(f_k\right)$ $\left(k=1,2,3,\cdot \cdot \cdot ,N\right)$ denote the discrete Fourier amplitude and phase, respectively.

Subsequently, Jin and Liao expressed $t_e$ based on the discrete recorded earthquake acceleration:

$$t_e(f)=-\frac{1}{2\pi }\frac{\Delta \phi (f)}{\Delta f}$$

(5)

where $\Delta \phi$ and $\Delta f$ are derived from discrete Fourier transform, $\Delta \phi \in [-2\pi ,0)$, $\Delta f=1/T$, $t_e\in (0,T]$, in which $T$ denotes the duration of strong ground motion.

It is observed from Equation (5) that when $\Delta \phi \in [-2\pi ,0)$, $t_e\in (0,T]$. In signal processing, the envelope delay $t_e$ describes the time shift of the envelope of a wave packet. Jin and Liao further demonstrated that $t_e\left(f\right)$ signifies the relative arrival time of a wave group with a frequency of $f$ corresponding to the initially arrived wave group for earthquake ground motion. In essence, the permissible range of $t_e$ should be within $(0,T]$ in physical terms. The method of calculating envelope delay using Equation (5) in which $\Delta \phi \in [-2\pi ,0)$ is known as the DFT method.

3.3. Case Study

Both techniques serve to compute the envelope delay of earthquake ground acceleration. To illustrate the range of envelope delays, two ground acceleration records with different durations, 6.85 s and 129.995 s, are considered. The detailed ground motion information is listed in

Table 2. Detailed information of the four accelerograms used in numerical cases of Section 3.

No.	Earthquake	Date	Station	Azimuth
1	Bishop (Rnd Val)	23 November 1984	McGee Creek-Surface	270°
2	El Mayor-Cucapah	4 April 2010	Chihuahua	0°

. The names of the earthquakes and stations are the same as those in the NGA West 2 database of the Pacific Earthquake Engineering Research Center (PEER). The accelerograms of the two ground motions are depicted in the top row of Figure 5.

Corresponding to the ground acceleration durations, the envelope delay ranges derived from DFT method are $[0,6.174 \mathrm{s}]$ and $[0,129.388 \mathrm{s}]$, respectively. As shown in Figure 5, the envelope delay calculated from the DFT method closely aligns with the time range of the ground motion time series, while results from the CFT method exhibit notable discrepancies. Moreover, the scatter diagram of envelope delay versus amplitude reveals that wave groups with larger amplitudes arrive during the period of strong shock, whereas those with smaller amplitudes arrive during the onset and attenuation periods. Scatter diagrams depicting envelope delay versus amplitude were created and analyzed using thousands of recorded earthquake ground accelerations. Consistently, the same conclusion was reached. This evidence indicates that the envelope delay calculated by the DFT method corresponds to the ‘relative arrival time’, facilitating the detection of non-stationarity in earthquake ground motions and the simulation thereof.

Additionally, as depicted in Figure 5, the envelope delay exhibits minimal dependence on frequency, suggesting challenges in directly modeling the phase spectrum. Based on the aforementioned analysis, the relationship between envelope delay and Fourier amplitude emerges as a pivotal breakthrough for simulating the phase spectrum.

4. Relationship between Phase Derivative and Duration

To further validate the temporal properties of the phase spectrum of earthquake ground motion, the relationship between phase derivative and duration was investigated. As observed in Section 3, all the displayed envelope delays were shorter than the total duration of the earthquake ground motion, with the maximum delay closely approaching the total duration. Hence, the total duration of the earthquake ground motion can be estimated solely from its phase spectrum:

$$\tilde{T}=\max \left[t_{\mathrm{e}}(\omega )\right]=\max \left[-\frac{\Delta \phi (\omega )}{\Delta \omega }\right]\hspace{1em}\hspace{1em}\Delta \phi (\omega )\in (-2\pi ,0]$$

(6)

where $\tilde{T}$ denotes the estimated total duration. The relationship between the duration and maximum envelope delay was illustrated by investigating 7778 earthquake ground motions recorded from 156 earthquakes. The range for moment magnitude, epicentral distance, and average shear wave velocity up to 30 m depth (v_s30) were 4.2–7.7, 0.44–675.91 km, and 89.3–2100 m/s, respectively.

The relative error of the largest envelope delay $\tilde{T}$ and the total duration $T$ was denoted by ${\delta }_T$, as shown in Equation (7). It was computed for all of the 7778 earthquake ground motions, with the resulting histogram displayed in Figure 6. It is shown that the majority of the ${\delta }_T$ s exhibit diminutive values, closely approaching zero. Additionally, the percentage of earthquake ground motions falling within specified relative error ranges $(-\infty ,-10\%]$, $(-10\%,-5\%]$ and $(-5\%,0]$ was calculated and is presented in

Table 3. Percentage of the number of earthquake ground motions with a certain range of ${\delta }_T$.

Range of Relative Errors	$(-\mathit{\infty },-10\%]$	$(-10\%,-5\%]$	$(-5\%,0]$
Percentage of the number of earthquake ground motions with a certain range of ${\delta }_T$ for the earthquake ground motions analyzed	1.11%	3.68%	95.22%

for the analyzed earthquake ground motions. As presented in Table 3, the relative errors, with an absolute value of less than 5%, constituted 95.22% of the total. Only 1.11% of all analyzed earthquake ground motions exhibit a ${\delta }_T$ below 10%. Therefore, Equation (6), as indicated in the literature [31], is applicable for estimating the total duration when the model of the phase spectrum is known.

$${\delta }_T=\frac{\tilde{T}-T}{T}$$

(7)

The maximum absolute value of ${\delta }_T$ was 33.3%. To probe the cause of the occurrence of large ${\delta }_T$ s, three typical accelerograms with the largest ${\delta }_T$ were analyzed, and their basic information is presented in

Table 4. Information of the typical earthquake ground motions with the largest ${\delta }_T\mathrm{s}$.

Earthquake	Date	Station Name/No.	$\mathit{T}$ (s)	$\tilde{\mathit{T}}$ (s)	${\mathit{\delta }}_{\mathit{T}}$
Northridge-01	17 January 1994	Pacoima Dam (downstr)	19.98	13.326	−33.3%
Coalinga-05	22 July 1983	Transmitter Hill	21.84	14.69	−32.7%
Victoria, Mexico	9 June 1980	Cerro Prieto	24.52	19.37	−21%

. The accelerograms and their scatter diagrams of the envelope delays and Fourier amplitudes are shown in Figure 7. The ${\delta }_T$ s of the three analyzed earthquake ground motions are −33.3%, −32.7%, and −21%, respectively.

As observed from Figure 7, despite the presence of a large ${\delta }_T$, the scatter diagrams of the envelope delays and Fourier amplitudes are in good agreement with the shape of the accelerogram. The wave groups with large amplitudes and strong shocks occurred simultaneously. Furthermore, a common feature among the three analyzed earthquake ground motions is evident: motions with very low amplitudes, approaching zero, persist for a significantly extended duration, nearly half of the duration. This observation suggests that the maximum envelope delay still remains closely aligned with the duration of motion exhibiting substantial amplitude.

The findings from Section 3 reveal a robust consistency between the shape of the scatter diagram, with the envelope delay represented on the horizontal axis and the Fourier amplitudes on the vertical axis, and the waveform. In Section 4, the results highlight a strong correlation between the envelope delays computed through the DFT method and the duration of the earthquake ground motion. Both sets of findings demonstrate the critical role of envelope delay in the time-domain non-stationarity of earthquake ground motions. In essence, the phase spectrum encapsulates the temporal characteristics and time-domain non-stationarity of earthquake ground motion.

5. Simulation of Strong Earthquake Ground Motions Based on the Envelope Delay

The correlation between envelope delay and Fourier amplitude offers a straightforward method for simulating strong ground motions. In this study, strong ground motion is treated as a time series fully described by its Fourier transform. Therefore, by specifying the Fourier amplitude spectrum $A\left(\omega \right)$ and phase spectrum $\phi \left(\omega \right)$, the ground motion $a(t)$ can be obtained using the discrete inverse Fourier transform.

$$a(t)={\displaystyle \sum _{k=1}^{N}A\left({\omega }_k\right)\cos \left({\omega }_{k}t+\phi \left({\omega }_k\right)\right)}$$

(8)

The Fourier amplitude spectrum of earthquake ground motion has been extensively studied and is insightfully represented by a physical random function model [32]. However, modeling the Fourier phase spectrum of ground motion remains a challenge due to its complexity. In this study, a new method based on the relationship between envelope delay and Fourier amplitude is proposed to represent the phase spectrum.

5.1. Fourier Amplitude Spectrum Model

The physical random function model of Fourier amplitudes comprises three fundamental components, as described by Equation (9) formulated by Wang and Li [32]: the source amplitude $A_s\left(\omega \right)$, the transfer function of path $H_{Ap}\left(\omega \right)$, and the transfer function of site $H_{As}\left(\omega \right)$. The source amplitude was derived from the Brune’s circular dislocation source model [23]. In this model, the earthquake’s rupture surface is assumed to be circular, with dislocation uniformly distributed across its surface. The transfer function of the path was established through an analysis of the damping dissipation energy within the propagation medium. This medium is considered to be a homogeneous elastic material. The transfer function of the site was obtained under the assumption that the site soil behaves as an equivalent single-degree-of-freedom system.

$$A\left(\omega \right)=A_s\left(\omega \right)\cdot H_{Ap}\left(\omega \right)\cdot H_{As}\left(\omega \right)$$

(9)

where

$$A_s\left(\omega \right)=\frac{\omega A_0}{\omega \sqrt{{\omega }^2+{\left(\frac{1}{\tau }\right)}^2}}$$

(10)

$$H_{Ap}=\exp (-K\omega x)$$

(11)

$$H_{As}=\sqrt{\frac{1+4{\xi }_g^2{(\omega /{\omega }_g)}^2}{{\left[1-{(\omega /{\omega }_g)}^2\right]}^2+4{\xi }_g^2{(\omega /{\omega }_g)}^2}}$$

(12)

in which $\omega$ denotes circular frequency; $x$ denotes the distance from the source to the local site; $A_0,\tau ,{\xi }_g,{\omega }_g$ denote the parameters of source intensity, Brune’s source, predominant damping ratio of the site, and predominant circular frequency of the site, respectively. These parameters serve as the fundamental random variables that represent the inherent randomness in the Fourier amplitude spectrum. The four basic random variables can be identified using earthquake ground acceleration records. The validity of the least squares method for the identification of $A_0,\tau ,{\xi }_g,{\omega }_g$ was demonstrated by Ding and Li [33].

5.2. Modeling of Fourier Phase Spectrum

The Fourier phase spectrum was generated in this study based on the relationship between envelope delay and Fourier amplitude. The envelope delays are uniformly distributed across various time ranges, determined by the shape of the scatter diagram depicting envelope delay and Fourier amplitude. Subsequently, phase angles are derived from the integration of envelope delay. This method requires only two parameters—namely strong shock duration and median arrival time—to simulate the phase spectrum.

Most earthquake ground motions typically feature only one strong shock, indicating that the scatter diagram between envelope delay and Fourier amplitude is unimodal, as illustrated in Figure 8, a record from the Chi-Chi earthquake. Therefore, a normal distribution curve can effectively represent the envelope of the scatter diagram. In Figure 8, $T_{ss}$ represents the strong shock duration or effective duration, while $t_m$ denotes the median arrival time of wave groups. For an actual ground motion time history, $T_{ss}$ is defined as the 5–95% duration during which 90% of the total energy is achieved. According to Parseval’s theorem, $T_{ss}$ is given by

$${\displaystyle \sum _{t_e=0}^{t_i}{A}^2(t_e)}={\displaystyle \sum _{t_e=t_j}^{\max (t_e)}{A}^2(t_e)}=5\%{\displaystyle \sum _{t_e=\min (t_e)}^{\max (t_e)}{A}^2(t_e)}$$

(13)

$$T_{ss}=t_j-t_i$$

(14)

in which $A(t_e)$ denotes the Fourier amplitude corresponding to the envelope delay $t_e$; $t_m$ is defined as the envelope delay corresponding to the largest amplitudes. To determine the value of $t_m$ from the recorded acceleration time history, the range of envelope delay is evenly divided into 100 intervals. The quadratic sum of amplitudes corresponding to each interval of envelope delay can then be readily calculated. The mean value of all the envelope delays within the interval associated with the maximum quadratic sum of amplitudes is denoted as $t_m$.

To reconstruct a strong ground motion time history, the normal distribution curve must first be derived. Three parameters are required: the mean value $\mu$, the standard deviation $\sigma$, and the peak value of the curve. The peak value of the curve represents the maximum amplitude and corresponds to the largest amplitude derived from the physical random function model. As the mean value $\mu$ represents the median envelope delay, it is given by

$$\mu =t_m$$

(15)

The strong shock duration $T_{ss}$, as defined by Equations (13) and (14), is significantly shorter than the total duration $T$ of the acceleration time history. Figure 9 depicts the statistical histogram of $T_{ss}/T$ for 7778 earthquake ground motions, with a mean value of this ratio calculated as 0.2336. In this study, $T_{ss}/T=0.25$ is utilized for calculating the Fourier phase. As illustrated in Figure 10, the probability densities of the standard normal distribution outside the range of $\left[\mu -2\sigma ,\mu +2\sigma \right]$ are relatively small. Therefore, it is assumed that $4\sigma =T$. Consequently, the standard deviation $\sigma$ of the normal distribution is given by

$$\sigma =\frac{1}{4}T=T_{ss}$$

(16)

With the normal distribution curve and model amplitudes established, each amplitude $A$ corresponds to two time points: $t_1$ and $t_2$, as shown in Figure 8. The envelope delay $t_e$ corresponding to this amplitude $A$ is uniformly distributed in $[t_1,t_2]$. Subsequently, the phase spectrum is calculated through the integration of the envelope delay $t_e$.

$$\phi ={\displaystyle \int -t_e}\mathrm{d}\omega$$

(17)

With both the amplitudes and phase angles having been determined, the time series of earthquake ground motion can now be generated using inverse Fourier transform.

5.3. Simulation of Ground Motion Accelerograms

For illustrative purposes, several tests were conducted to evaluate the goodness of fit of simulated ground motions to recorded ones. These tests are crucial for validating the accuracy and reliability of the simulation method. Two earthquake ground acceleration records were investigated in this study, each exhibiting distinct characteristics in terms of duration and spectral components. The first record is from the Chi-Chi earthquake, which displays a strong shock duration of 17.47 s and a median arrival time of 36.16 s. In contrast, the record from Northridge-03 earthquake demonstrates a much shorter strong shock duration of 3.91 s and a median arrival time of 6.15 s. These differences in duration and spectral components highlight the diverse nature of seismic events and underscore the importance of accurately capturing such variations in simulation models.

The two accelerograms, along with their simulations generated based on the proposed simulation method in this study, are depicted in Figure 11. The comparison illustrates that the simulated ground motion acceleration exhibits similar time-domain non-stationarity to the recorded data. Furthermore, Figure 12 showcases the acceleration response spectra of both the recorded data and 10 simulations. It is apparent from the figure that the simulated samples closely align with the records in terms of spectral properties, thereby further validating the accuracy of the simulation method.

6. Discussion

The frequency-dependent phase derivative, along with the relationship between phase derivatives and amplitudes, plays a significant role in the non-stationarity of earthquake ground motions. The envelope delays calculated using the DFT method serve to substantiate the physical significance of the phase derivative and elucidate the relationship between the phase spectrum and the amplitude spectrum. The value of the envelope delay calculated using the DFT method aligns perfectly with the duration of the ground motion. Although envelope delay has been considered in earthquake engineering for some time [20], previous computation methods yielded envelope delays that were significantly different from the duration of the earthquake ground motion. This study offers a substantial improvement over previous computation methods, thereby clarifying the physical significance of the phase spectrum and applying the relationship between phase spectrum and amplitude spectrum to simulate earthquake ground motions.

Compared to the traditional and well-known spectral representation method for simulating earthquake ground motions [4,6], the proposed approach offers a novel and more efficient way to simulate non-stationary earthquake ground motions by leveraging the relationship between the amplitude spectrum and the phase spectrum. Both methods use the same number of random variables, corresponding to the number of amplitudes. However, in the spectral representation method, the temporal non-stationarity of earthquake ground motions is described by a temporal modulation function, requiring at least three parameters. In contrast, the proposed method describes the temporal non-stationarity using only the phase angles and the relationship between the phase spectrum and amplitude spectrum, reducing the requirement to just two parameters with clear physical meaning: $t_m$ and $T_{ss}$.

In future work, $t_m$ and $T_{ss}$ will be identified based on recorded earthquake ground motions. The probability density functions of $t_m$ and $T_{ss}$ will then be determined from a large dataset of recorded earthquake ground motions. By reducing the number of parameters from three to two, the simulation of earthquake ground motions is simplified. The proposed method enables the simulation of a random set of ground motions that meet established statistical properties.

7. Conclusions

The simulation of seismic ground motions plays an important role in assessing the impact of earthquakes on buildings. By replicating the complex dynamics of ground shaking, engineers and researchers can predict how various structures will respond to seismic events. This involves analyzing the stress, strain, and potential failure points in buildings, allowing for the design of more resilient infrastructure. Such simulations are essential for improving building codes and standards, ultimately enhancing the safety and durability of urban environments in earthquake-prone regions.

The simulation method proposed for earthquake ground motions represents a notable advancement in accurately replicating the primary features observed in strong ground motion recordings. Its efficacy lies in its ability to achieve this fidelity while operating with a minimal set of parameters necessary for phase description. This streamlined approach not only simplifies the simulation process but also enhances its practical applicability. A significant advantage of this method is its capability to derive phase information directly from amplitude data, relying on just two parameters. Each of these parameters carries distinct physical significance, facilitating a straightforward interpretation of the simulated ground motion characteristics. This inherent simplicity not only expedites the simulation process but also enhances the method’s accessibility to a broader range of users, including researchers and practitioners in earthquake engineering and seismology. While the determination of parameter values remains empirical, the method nonetheless offers a rapid simulation of acceleration time histories. This makes it suitable for engineering applications across a wide range of magnitudes, distances, and site types. Furthermore, the probability density functions of the parameters can be estimated from the amount of earthquake ground motion recordings. Thus, the uncertainty associated with the ground motions can be easily and quickly revealed.

In certain earthquake events, such as the Wenchuan earthquake, two strong shocks are observed in a single ground motion due to multiple fault ruptures and extended rupture intervals. The simulation method proposed in this study cannot simulate these types of ground motions because it uses a unimodal normal distribution curve to describe the amplitude–phase relationship. Future research will address this limitation to broaden the applicability of the proposed simulation method.