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Near-fault ground motions exhibit significant characteristics such as velocity pulses, rupture directivity, and strong vertical components, which pose serious threats to structural safety. However, near-fault ground motion records remain scarce and have not been adequately accounted for in current seismic design codes. This paper proposes a data-driven simulation method for non-stationary near-fault ground motions based on Stationary Wavelet Transform (SWT) combined with Hilbert’s instantaneous frequency estimation. First, to address the baseline drift issue commonly observed in measured seismic motions, a baseline correction technique combining the least squares method and the Iwan method is proposed to enhance the reliability of seismic time histories. Subsequently, statistical distributions of velocity pulses and vertical-to-horizontal (V/H) acceleration ratios, along with their relationships with fault distance and magnitude, are analyzed based on more than 900 ground motion records. The results show that these near-fault motions generally contain pronounced long-period components, which will have significant implications for the seismic response of long-period structures. Additionally, unidirectional pulses dominate in near-fault records. Among the 107 selected long-period pulse records, unidirectional pulses account for 69.2%. Based on this, seismic motions are decomposed using SWT, and stochastic reconstruction is performed, combined with multivariate response spectrum matching to optimize the generation of near-fault time histories consistent with the target spectrum. Compared with the results obtained without optimization, the proposed method reduces the mean square error by about 40% or more, demonstrating a clear improvement in accuracy and reliability. This method provides reliable seismic input support for seismic analysis and performance-based design of bridges in near-fault regions.

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1. Introduction

In recent years, a number of destructive earthquakes have caused severe damage to buildings and urban infrastructure. For example, the 2023 Kahramanmaraş earthquakes in Türkiye devastated 11 provinces and impacted more than 10 million people, with building collapse rates reaching nearly 15% in the most affected areas [1], while the 2025 Balıkesir earthquake once again resulted in extensive damage across the affected areas [2]. Research has shown that the severity of structural damage is often governed by the characteristics of strong ground motions occurring near fault rupture zones [3]. Near-fault ground motions, characterized by pulse-like velocity records and strong directivity effects, can impose significantly higher demands on structures compared with ordinary far-fault motions [4]. Therefore, understanding and accurately simulating near-fault seismic motions is essential for the development of reliable performance-based seismic design frameworks. Somerville et al. [5] pointed out that near-fault ground motions, due to rupture directivity effects, often generate distinct pulse-like velocity records that result in concentrated energy input to structures. Kalkan et al. [6] found that pulse-type near-fault ground motions can significantly amplify structural displacement and strength demands compared with ordinary (far-field or non-pulse) records. Güneş [7] further investigated the influence of pulse-like ground motions on seismically isolated buildings and demonstrated that near-fault excitations can notably increase isolator displacements and structural responses compared with far-fault cases. Zhang et al. [8] reported that, under near-fault ground motions, the maximum base shear of structures can reach up to 5.10 times that of far-field motions and 1.47 times that of non-pulse near-fault motions. More recently, Liu et al. [9] analyzed single-pulse-like and double-pulse-like characteristics of near-fault records, showing that the variation in pulse type can significantly affect the dynamic response and energy input mechanisms of structures. Similarly, Zhai et al. [10] investigated the seismic performance and failure modes of long-span continuous rigid-frame highway bridges under near-fault ground motions by comparing them with far-field cases. The results indicated that near-fault motions with pulse effects produce much more intense dynamic responses than far-field motions. Currently, there is no unified definition of the near-fault region internationally, though many scholars consider the area within 20 km of the fault rupture surface as the near-fault zone [11]. To clarify the relationship between ground motion characteristics and distance, researchers have proposed various attenuation relationships to describe the decay of seismic motion with distance. In general, attenuation includes two main components: geometric attenuation, which reflects the decrease in amplitude due to geometric spreading, and non-elastic (anelastic) attenuation, which varies with regional geological and tectonic conditions and represents additional energy loss caused by material absorption and scattering. Specifically, numerous empirical attenuation (ground motion prediction) models have been developed based on different datasets and regional conditions. Boore et al. [12] developed an empirical model applicable to active crustal regions in North America using the NGA-West1 database and multivariate regression analysis of ground motion parameters. Akkar et al. [13] established empirical ground motion prediction equations (GMPEs) for shallow crustal earthquakes in the Mediterranean and Middle Eastern regions using the European Strong-Motion Database and additional regional data. Similarly, Campbell et al. [14] proposed GMPEs suitable for active crustal regions in western North America based on the NGA-West2 strong-motion database. Due to differences in geological conditions, magnitude ranges, and regional datasets used, these models differ in data composition, regional applicability, and functional form [15]. Therefore, attenuation relationships are not universally applicable; careful selection or modification is required, particularly when modeling near-fault ground motions. Bray [16] and Mavroeidis [17] studied the propagation and attenuation characteristics of near-fault ground motions using seismic data within 20 km of the fault. They extended known concepts by comparing these motions with far-field ground motions. Many researchers have analyzed structural responses under near-fault ground motions to clarify how the velocity-pulse characteristics influence structural damage. For instance, Shi et al. [18] examined the seismic performance of a high-speed railway bridge under various near-fault excitations and found that pulse-type ground motions could induce excessive deformation in energy-dissipating components, highlighting the strong sensitivity of structures to pulse effects. Early work by Bertero [19] established the fundamental relationship between structural strength, deformation capacity, and seismic demand, providing the theoretical framework for evaluating damage under severe ground motions. Building on these foundations, Somerville [20] developed an improved representation of near-fault ground motions, identifying key velocity-pulse and directivity effects that distinguish them from ordinary far-fault motions. Somerville [21] further emphasized that these near-fault characteristics should be incorporated into seismic hazard assessment, as forward-directivity effects can significantly alter ground motion intensity and duration parameters. Abrahamson [22], through the analysis of near-fault recordings from the 1999 Chi-Chi earthquake, provided empirical evidence confirming the occurrence of strong velocity pulses and their pronounced influence on structural response. Collectively, these studies demonstrate that analyzing structural responses under near-fault conditions helps to establish the relationship between velocity-pulse characteristics and structural damage, thereby contributing to a more refined understanding of near-fault seismic behavior.

The spectral characteristics of near-fault ground motions are a cutting-edge topic in structural seismic analysis. Xing et al. [23] applied wavelet analysis to study the intensity indices of near-fault ground motions and their value ranges and investigated the effects of near-fault ground motions on structural seismic response and dynamic stability. Additionally, long-period velocity pulses in near-fault ground motions can cause larger seismic displacements and deformations in long-span bridges, warranting significant attention to their impact on such structures. Xu [24] and Wei [25] found that velocity pulses in near-fault ground motions are closely linked to the fault rupture direction and the occurrence of permanent displacements. Moreover, near-fault refers to areas near the fault rupture surface (ranging from tens to hundreds of kilometers), while near-field refers to regions close to the seismic source.

Near-fault ground motions differ significantly from far-field motions, attracting extensive research both domestically and internationally. Du et al. [26] used a three-dimensional finite difference fault transient rupture dynamics model, provided by Dr. Steven Day of the Southern California Earthquake Center (SCEC), to study the dynamic rupture process of the 1976 Tangshan earthquake fault and the characteristics of near-fault strong ground motions. They also discussed the impact of seismic rupture directionality on near-fault surface motions. Xie et al. [27] analyzed 64 strong-motion records from faults within 200 km of the Wenchuan earthquake using wavelet methods. They found that the near-fault ground motions exhibited long periods (6–14 s) and small amplitudes, with all records located within 30 km of the surface rupture. Xu et al. [24] studied the three-component near-fault ground motions of the Chi-Chi earthquake in Taiwan, identifying the fault rupture directionality and the hanging wall effect of reverse faults as prominent features of near-fault seismic motions. Wei et al. [25] investigated the seismic characteristics of near-fault velocity pulses and found significant differences between the response spectra of near-fault ground motions and the standard spectra in China’s seismic design code, especially for pulse-containing motions, which exhibited notably higher spectra. For characteristic periods, the response spectra T1 and T2 of near-fault motions showed a delayed trend. Bouchon et al. [28] analyzed the 1999 Turkish earthquake waves using near-fault acceleration plots to infer the spatial–temporal rupture process of the fault, finding that seismic velocity and displacement near the rupture zone exhibited simple pulse forms. Mavroeidis GP [17] proposed a simple and effective numerical model for simulating near-fault seismic motions, which qualitatively and quantitatively described their pulse characteristics. Arben Pitarka et al. [29] examined artificial near-fault ground motions from the 1995 Kobe earthquake using a three-dimensional moving fault model and a simplified three-dimensional velocity structure, applying the finite difference method (FDM) to model seismic rupture and wave propagation. The results highlighted the significant role of the coupling between basin edge effects and seismic source directionality in amplifying seismic motions, resulting in very high-amplitude pulse-like ground motions.

Ground motion selection critically governs structural seismic response; different inputs can produce markedly different demands. Prior near-fault studies often rely on a few “representative” records for dynamic analysis, with limited attention to how specific near-fault features affect response. Velocity pulses are a principal driver of structural damage in near-fault areas. Tian et al. [30] analyzed and summarized 28 near-fault ground motion records containing velocity pulses and found that the pulse amplitude and ground motion characteristics have a significant influence on the response spectra of near-fault earthquakes. Similarly, Fäcke et al. [31] applied near-fault ground motions considering the source mechanism and site effects in bridge analyses and found that incorporating these factors led to a significant increase in the bridge response. Moreover, although design practice typically emphasizes horizontal components, near-fault vertical accelerations can exceed their far-field counterparts and are non-negligible for structural performance. These observations motivate dedicated study of near-fault ground motions to address design needs. To address the scarcity of recorded near-fault ground motions, it is urgently necessary to develop an effective simulation approach. Obtaining accurate ground motion records is a prerequisite for such studies. However, due to instrumental tilts or long-period noise, the collected seismic data often contain baseline drifts, which distort the true characteristics of the ground motion. Boore [32] analyzed recordings from the 1999 Chi-Chi earthquake and proposed a polynomial and segmental fitting method to remove low-frequency trends, demonstrating its effectiveness in reducing artificial permanent displacements. Graizer [33] developed an automatic empirical baseline correction algorithm that detects and eliminates both linear and nonlinear drifts without manual intervention, improving the stability of long-duration and noisy records. Chen et al. [34] introduced a deep neural network model based on empirical mode decomposition to solve baseline correction by removing the drifting trend, which effectively suppresses residual drifts. However, these approaches still face limitations when dealing with strong near-fault or pulse-like ground motions, where nonlinear effects and complex low-frequency components are significant. To overcome this limitation, this study proposes a baseline drift correction technique that combines the least-squares method with the Iwan model, which demonstrates excellent performance for near-fault ground motions.

Regarding ground motion simulation, Yamamoto et al. [35] employed the wavelet packet transform (WPT) to decompose earthquake records into multiple narrow-band components and proposed a stochastic reconstruction approach to generate non-stationary artificial ground motions with realistic time–frequency characteristics. Wen et al. [36] further extended this idea by developing a wavelet-packet-based multivariate simulation framework, which can effectively capture the time-varying spectral correlation among multiple components of ground motion. In addition, Liu et al. [37] utilized the ensemble empirical mode decomposition (EEMD) method to analyze near-fault ground motions and demonstrated that empirical decomposition can extract pulse-like and nonstationary features suitable for modeling and simulation. Although these methods have significantly improved the realism of artificial ground motions, they still exhibit some limitations. WPT-based methods lack translation invariance, which can cause information loss or signal distortion during reconstruction, while EEMD lacks a rigorous mathematical foundation and depends heavily on empirical parameter selection. To overcome these drawbacks, this study proposes an improved near-fault ground motion simulation approach that integrates the stationary wavelet transform (SWT) and Hilbert analysis, in which a non-Newtonian linear optimization algorithm is employed to enhance the accuracy and stability of the artificial motions.

2. Data and Methods for Near-Fault Ground Motion Analysis

2.1. Near-Fault Ground Motion Characteristics

Numerous studies have shown that near-fault ground motions exhibit distinct characteristics such as short-duration high-energy pulse motion, fault rupture directionality effects, strike–slip effects, reverse fault earthquake hanging wall effects, and large vertical accelerations, all of which significantly differ from far-field seismic motions. The specific characteristics of near-fault ground motions are summarized as follows:

(1). Compared with far-field ground motions, near-fault ground motions exhibit a significantly narrower velocity-sensitive region, while their acceleration and displacement-sensitive regions are considerably broader. This characteristic indicates that the energy of near-fault ground motions is mainly concentrated in the longer-period range, resulting in long-period dominant motions. The primary cause of this phenomenon lies in the rupture directivity and velocity pulse effects of near-fault earthquakes, which concentrate the seismic energy release within a short duration and amplify the long-period components of ground motion [6].

(2). Near-fault ground motions feature strong vertical components concurrent with large horizontal shaking. Peak horizontal accelerations are high and strongly anisotropic, differing markedly between the fault-normal and fault-parallel components, with a pronounced hanging-wall effect. With increasing distance from the fault, peak accelerations attenuate rapidly, and response-spectrum ordinates decrease sharply with period [38,39]. Large vertical components are documented in many near-fault strong-motion records: the 1971 San Fernando earthquake yielded a vertical peak ground acceleration (PGA) of 1.25 g (then among the largest observations); the 1979 Imperial Valley earthquake recorded a vertical PGA of 1.655 g at a site approximately 1 km from the rupture, which was 3.77 times the corresponding horizontal peak; and the 2008 Wenchuan earthquake exhibited a vertical-to-horizontal (V/H) PGA ratio of 1.413 [40]. Such ratios substantially exceed the 1/2–2/3 range adopted in many national codes. Analyses of near-fault strong-motion records indicate that attenuation of vertical PGA is more accurately characterized when fault distance and magnitude are jointly considered. For many strong events within 0–40 km of the fault, V/H PGA ratios commonly surpass the code value of two-thirds. Conversely, pronounced velocity pulses in the near-fault region can reduce these ratios despite large acceleration peaks.

(3). Near-fault ground motions are strongly controlled by rupture directivity (Figure 1). When the rupture front propagates toward a site with high apparent velocity, recordings often show forward-directivity pulses. These pronounced long-period pulses in the velocity time history are particularly damaging to medium- and long-period structures.

(4). Velocity pulses are a defining characteristic of near-fault ground motions. They arise from two mechanisms: forward rupture directivity and permanent ground displacement. The relationship between pulse orientation and faulting for strike–slip and dip–slip cases is illustrated in Figure 2. Two archetypal forms are observed [41]. Unidirectional pulses refer to ground motions that exhibit a distinct velocity pulse in only one horizontal component, primarily caused by permanent ground displacement, and they typically dominate the component parallel to the fault slip direction. Bidirectional pulses, by contrast, refer to ground motions in which significant velocity pulses occur simultaneously in both horizontal components, produced by rupture-propagation directivity (a Doppler-like effect) and sometimes manifesting as multi-lobe pulses, typically appearing in the fault-normal direction. These large-amplitude, long-period velocity pulses can induce excessive lateral displacement and residual deformation in structures whose natural periods are close to the pulse period, thereby causing permanent damage.

(5). Hanging-wall effects are pronounced in earthquakes on dipping faults, producing systematic contrasts between hanging-wall and footwall ground motions. These effects modify near-fault peak amplitudes, response spectra, durations, and spatial gradients. Observations indicate that within roughly 30 km of the fault, response spectra on the hanging-wall exceed those on the footwall over a broad period range [42,43,44,45]. The amplification arises chiefly because hanging-wall sites are geometrically closer to the rupture plane in reverse or thrust events.

2.2. Near-Fault Record Selection and Baseline Correction

2.2.1. Requirement for Baseline Correction of Near-Fault Strong-Motion Records

Although there are many active faults in Southwest China and numerous major earthquakes have occurred in history, there are almost no records of near-fault seismic acceleration in this region. Therefore, this study can only analyze near-fault seismic records from other regions to explore the common characteristics of near-fault ground motions and use these characteristics as the theoretical basis for seismic studies of bridges in near-fault areas of Southwest China.

To facilitate a comparative analysis between near-fault and far-field ground motion characteristics, seismic records with moment magnitude (Mw) ≥ 5.0 and peak ground acceleration (PGA) ≥ 0.1 g were selected. The dataset includes events with rupture distances within 90 km. The rupture distance is defined as the shortest distance from the recording station to the fault rupture plane, as shown in Figure 2. In total, 975 horizontal and 318 vertical components were collected from domestic and international networks, among which 408 records with fault distances ≤ 20 km were used specifically for the investigation of near-fault ground motion characteristics. A subset of representative metadata for the selected ground motion records is summarized in Table 1.

In the above near-fault earthquake records, many of them exhibit baseline drift. Accordingly, baseline correction should precede any characterization of these near-fault ground motions. Prior studies [17,28,46] attribute baseline drift to instrument-response inaccuracies, electromagnetic interference, sensor aging, material fatigue, and permanent displacements of the instrument base during seismic motion, etc. The main causes are summarized as follows:

(1). Low-frequency errors

Digital strong-motion recordings contain instrument noise and ambient site noise. Instrument noise primarily stems from electronic interference, inadequate sampling rate or resolution, and minor hysteresis in sensor materials and circuits; for example, electronic noise can produce slight zero-baseline fluctuations in the recorded signal. Ambient site noise arises from environmental vibrations (e.g., waves and wind). Despite the diversity of sources and frequency content, the contribution of any single source is usually small, so the aggregate is commonly modeled as Gaussian white noise; nevertheless, its amplitude is strongly influenced by local site conditions [47].

(2). Instrument tilt

Nonuniform ground deformation in near-fault regions can tilt the strong-motion recorder, and even small tilts can induce pronounced baseline (zero-line) drift. The direct impact on the recorded acceleration is typically minor, usually less than 2% of the peak ground acceleration (PGA), but time integration amplifies this bias, leading to appreciable errors in velocity and even larger errors in displacement. As illustrated in Figure 3, when the sensor tilts by an angle θ, gravity projects onto the spring (sensing) direction with a component Mgsinθ (where M is the proof mass and g is gravitational acceleration). For inertial (proof-mass) accelerometers, this projection induces an apparent acceleration of gsinθ. Observations from near-field records indicate that this tilt-induced error primarily affects the uncorrected horizontal components of the velocity and displacement time histories, while its effect on the vertical component remains minor [48,49].

2.2.2. Baseline Correction Methods for Near-Fault Ground Motion Records

Extensive studies have proposed various baseline correction schemes for strong-motion records. Because different methods adopt different assumptions about the causes of baseline drift, the corrected near-fault results may differ across approaches. Two categories that are widely used are summarized below [50,51,52].

(1). Filtering algorithms targeting low-frequency errors

The U.S. Geological Survey’s Basic Acceleration Processing (BAP) software follows a straightforward idea: fit a best-fit straight line to the acceleration baseline, subtract it from the raw time history, and then filter the residual. A practical workflow is:

Step 1 (baseline initialization): Subtract the pre-event mean of the acceleration over 0–20 s from the entire record.

Step 2 (high-pass filtering): Remove low-frequency content using a Butterworth high-pass filter. For simplified processing, FFT-based or pseudo-FFT approaches can also be used to suppress low frequencies.

Step 3 (time integration): Integrate the corrected acceleration to obtain the velocity and displacement time histories.

(2). Correction for instrument tilt

In practice, baseline drift can arise from multiple mechanisms, and the evolution of instrument tilt during shaking is highly uncertain. Field observations indicate that, once tilt occurs, the slope of the baseline (i.e., the offset of the acceleration baseline) is approximately constant and can be represented by a straight line. Accordingly, it is commonly assumed that the strong-motion instrument experiences an instantaneous tilt and then maintains a constant tilt angle. The correction steps are:

Step 1 (baseline initialization): Subtract the pre-event mean of the acceleration over 0–20 s (denoted RAP) from the entire record, and integrate once to obtain the velocity time history.

Step 2 (terminal-segment fit): Fit the tail of the velocity record with a straight line,

(1) $v_{f} (t) = v_{0} + a_{f} t$

to obtain the velocity-baseline slope

a_{f}

and the intercept time

t_{w}

. Subtract

a_{f}

from the acceleration record for

t \geq t_{w}

Step 3: Integrate the adjusted acceleration to obtain velocity, subtract the pre-event mean over 0–20 s, and set the initial velocity to zero.

Step 4: Integrate once more to obtain the displacement time history.

Both methods are suitable for near-fault strong motions with large amplitudes. Nevertheless, numerical results show that displacement-baseline drift can remain significant after correction. The primary reason is that different assumptions about the drift mechanism lead to different acceleration baselines. At present, there is no universally accepted procedure for baseline correction of digital strong-motion records; case-specific adjustments are often required.

To further reduce residual drift, we propose an least-squares (LS)-based baseline fit guided by two principles. Firstly, corrected samples should be distributed as evenly as possible about a zero-mean line. Secondly, the polynomial order of the baseline model should be kept as low as possible. After iterative testing, the displacement baseline is satisfactorily represented by Equation (2):

(2) $u_{g} (t) = a_{1} t^{4} + a_{2} t^{3} + a_{3} t^{2} + a_{4} t$

where

a_{1}

a_{4}

are unknown coefficients.

Differentiating Equation (2) yields the velocity and acceleration baselines as follows:

(3) $u_{g}^{'} (t) = 4 a_{1} t^{3} + 3 a_{2} t^{2} + 2 a_{3} t + a_{4}$

(4) $u_{g}^{″} (t) = 12 a_{1} t^{2} + 6 a_{2} t + 2 a_{3}$

For the velocity baseline, the initial value is commonly taken as zero, so $u_{g}^{'} (0) = 0$ implies $a_{4}$ = 0 in Equation (3). The coefficients in Equation (4) are obtained by LS fitting to the measured acceleration $u_{g}^{″} (t)$ . For discrete samples $u_{g i}^{″}$ (i = 1, …, N), the LS objective is defined by Equation (5):

(5) $M i n \{[\sum_{i = 1}^{N} {(u_{g i}^{″} - u_{g}^{″})}^{2}]\} = M i n \{[\sum_{i = 1}^{N} [{(u_{g i}^{″} - (12 a_{1} t^{2} + 6 a_{2} t + 2 a_{3}))}^{2}]]\}$

and the normal equations are as follows:

(6) $\frac{\partial}{\partial a_{k}} \{[\sum_{i = 1}^{N} [{(u_{g i}^{″} - (12 a_{1} t^{2} + 6 a_{2} t + 2 a_{3}))}^{2}]]\} = 0$

The corrected displacement is then renewed by Equation (7):

(7) $\bar{u_{g}} = u_{g} (t) - u_{g}^{″} (t)$

While LS fitting can substantially alleviate drift, very long-period residual trends may persist. Therefore, we combine the LS approach with the Iwan method to construct a displacement-baseline elimination algorithm (Figure 4).

Applying the proposed method to near-fault acceleration records yields satisfactory results. Figure 5 compares raw and corrected acceleration, velocity, and displacement for the Upland earthquake recorded at the Rancho Cucamonga-FF station (90° component).

2.3. Long-Period Velocity-Pulse Characteristics of Near-Fault Ground Motions

Velocity pulses are a defining feature of near-fault ground motions [9,53,54]. The first pulse was reported in the 1966 Parkfield earthquake [55]; the first clearly damaging pulse was identified in the 1971 San Fernando event [56]. Numerous near-fault records from the 1999 İzmit and Chi-Chi earthquakes and the 2008 Wenchuan earthquake also exhibit pronounced pulses [57]. Because there is currently no universally accepted criterion for pulse identification, this study defines ground motions satisfying the following three conditions as pulse-like ground motions [58]:

(1). The velocity time history contains a sharp excursion (Figure 6).

(2). The excursion persists T_p for at least 0.5 s.

(3). The largest peak is at least twice the second-largest peak; if the two leading peaks are comparable, the smaller of the two is at least twice any other local maximum.

In general, velocity pulses have large amplitudes and long periods but short durations. Two primary mechanisms are recognized:

(1). In the fault-normal direction, a forward-directivity (Doppler-like) effect from rupture propagation produces a bidirectional pulse. Pulse amplitude diminishes with increasing rupture distance, indicating occurrence within a limited near-fault zone.

(2). In the fault-parallel direction, permanent ground displacement produces a unidirectional pulse.

Using the compiled near-fault dataset, acceleration records were integrated to velocity. Not all near-fault records are pulse-like. Among those that are, pulses may be confined to the horizontal component only or to the vertical component only. As summarized in Figure 7, only 9.56% of near-fault records contain pulses overall; among horizontal records, the fraction is 9.86%, and among vertical records, it is 8.34%. For 107 records with long-period pulses (Figure 8), unidirectional pulses account for 69.2% and bidirectional pulses for 30.8%. Within horizontal pulses, 79.7% are unidirectional and 20.3% bidirectional; within vertical pulses, 78.9% are unidirectional and 21.1% bidirectional. Overall, unidirectional pulses predominate.

Existing studies [28] implicate near-fault velocity pulses as a primary driver of bridge damage. Establishing how key pulse parameters relate to seismological attributes is therefore essential for assessing structural demand. Earlier research has demonstrated that peak ground acceleration (PGA), peak ground velocity (PGV), and the characteristic period (T) are representative parameters used to describe the intensity and frequency characteristics of ground motions [59]. PGA reflects the overall strength of ground shaking, PGV is related to the energy content of the seismic motion, T denotes the pulse period, which corresponds to the dominant frequency component of the velocity pulse and serves as a key indicator of its duration and energy content. In addition, pulse-type motions are characterized by large peak ground velocity (PGV) and a distinct pulse period T. Several studies [28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,60,61] report that large PGV/PGA ratios tend to coincide with substantial permanent ground displacements. However, the global inventory of recorded pulse-like motions remains limited, yielding scattered statistical trends. Moreover, many prior analyses did not explicitly consider the effects of faulting mechanism, fault burial depth, or rupture velocity on pulse characteristics. Accordingly, 107 pulse-type records identified in Figure 9 (74 unidirectional and 33 bidirectional) were analyzed to quantify relationships between the principal pulse parameters (T, PGA, PGV, PGV/PGA) and site attributes (rupture distance, moment magnitude, shear wave velocity Vs₃₀), as well as the dependence of PGA on T.

Figure 9 presents the relationships between the pulse parameters and rupture distance: (1) The pulse period T tends to decrease with increasing rupture distance; the maximum observed period is 5.0 s (Figure 9a). (2) PGA generally decreases as rupture distance increases (Figure 9b). (3) PGV decreases with rupture distance, with the trend most evident within approximately 10 km of the fault (Figure 9c). (4) The PGV/PGA ratio also decreases with rupture distance, and this trend is likewise most pronounced within about 10 km (Figure 9d).

Figure 10 presents the relationships between the pulse parameters and earthquake magnitude (Richter scale, used here as a proxy for released energy): (1) The pulse period T generally increases with magnitude, most clearly for Mw 6–8; unidirectional pulses vary markedly for Mw 6.5–8, whereas bidirectional pulses vary mainly for Mw 6–6.5; unidirectional pulses are broadly distributed over Mw 5.7–7.6, and bidirectional pulses over Mw 4–8 (Figure 10a). (2) PGA first increases and then decreases with magnitude, peaking near Mw 6.7 (Figure 10b). (3) PGV increases with magnitude, with a slightly larger rate for unidirectional than for bidirectional pulses (Figure 10c). (4) PGV/PGA increases with magnitude (Figure 10d).

Figure 11 presents the relationships between the pulse parameters and site condition Vs₃₀ (the time-averaged shear-wave velocity in the upper 30 m, used in Euro-American codes): (1) Pulse records occur predominantly for 200 m/s < Vs₃₀ < 700 m/s; bidirectional pulse periods show no clear trend with Vs₃₀, whereas unidirectional pulse periods increase and then decrease, peaking near Vs₃₀ = 500 m/s (Figure 11a). (2) PGA is concentrated with the scope of 200 m/s < Vs₃₀ < 700 m/s; for Vs₃₀ < 500m/s, both unidirectional and bidirectional pulses have maximum PGA around 0.8 g, while for Vs₃₀ > 500 m/s, unidirectional-pulse maximum PGA is around 0.4 g (Figure 11b). (3) PGV is likewise concentrated within the scope of 200 m/s < Vs₃₀ < 700 m/s, with larger amplitudes clustering near Vs₃₀ of approximately 500 m/s (Figure 11c). (4) For Vs₃₀ < 500m/s, PGV/PGA is below 300 s/100; the ratio rises sharply near Vs₃₀ = 500 m/s and then decreases with further increases in Vs₃₀ (Figure 11d).

Figure 12 presents the relationships between pulse PGA and pulse period T: (1) PGA varies non-monotonically with T. PGA decreases with increasing T at first, rises slightly to a maximum of approximately 1.2 g at T = 0.5~1 s, and then decays; for T > 1.0 s, PGA generally remains below 0.8 g, and the occurrence of pulse records diminishes as T increases (Figure 12a). (2) Horizontal pulses are mainly distributed over T = 0.5~5 s (Figure 12b). (3) Vertical pulses are mainly distributed over T = 0.5~3 s (Figure 12c).

The above analyses indicate that near-fault ground motions are highly complex and influenced by many factors. To isolate the salient pulse characteristics, an equivalent velocity-pulse component should be extracted numerically from recorded motions. This separation provides a clean basis for subsequent assessment of structural response under pulse-type excitation.

Wavelet analysis is well suited to nonstationary time histories and therefore appropriate for seismic acceleration records. In near-fault applications, wavelet decomposition can isolate the dominant long-period velocity pulse. A Daubechies wavelet is employed because its waveform resembles recorded pulses and it exhibits robust performance in pulse extraction. The procedure is:

(1). Integrate the recorded ground-acceleration time history to obtain the ground-velocity record.

(2). Apply a one-dimensional continuous wavelet transform (CWT) using the fourth-order Daubechies wavelet (Figure 13) and identify the maximum wavelet coefficient.

(3). Reconstruct the pulse component associated with this maximum coefficient and subtract it from the original velocity record to obtain a residual velocity record. The corresponding acceleration time histories can then be obtained by differentiating the velocity record.

(4). Repeat the CWT on the residual velocity, iteratively subtracting the component corresponding to the current maximum coefficient, until a pulse-free residual velocity record is obtained.

(5). Superpose the extracted components to form the composite velocity-pulse signal for the event.

Using the above wavelet procedure, the long-period velocity pulse was isolated from integrated acceleration records, and acceleration time histories with the pulse component removed were computed. Pulse extraction was carried out for multiple mainshock area records. Representative results are shown in Figure 14.

2.4. Vertical-Component Characteristics of Near-Fault Ground Motions

Damage surveys indicate that in high-intensity regions, particularly in epicentral and near-fault zones, vertical shaking is pronounced and may exceed the horizontal component. Given the potential for severe structural damage, detailed characterization of the vertical component in near-fault ground motions is warranted [62].

Many national codes require the vertical component to be considered for long-span and super-tall structures, typically by scaling the vertical action as a fraction of the horizontal (V/H). A common assumption is that the vertical PGA is one-half to two-thirds of the horizontal PGA. However, several major earthquakes in recent decades have recorded very strong vertical motions, in some cases larger than the horizontal. Accordingly, for bridges located in near-fault regions, the vertical design spectrum should not be taken simply as two-thirds of the horizontal spectrum in the current Railway Seismic Design Code; instead, the characteristic period and related parameters should be selected with explicit consideration of fault type, rupture distance, and other source-to-site factors [63,64].

Based on global records with magnitude greater than Mw 5, PGA exceeding 0.1 g, and rupture distance less than 20 km, near-fault motions were compiled and the distribution of V/H was obtained (Figure 15). The proportions are: V/H > 2/3 in 45.98%, V/H > 1.1 in 8.43%, V/H > 1.6 in 3.45%, and V/H > 2.1 in 1.15%.

Figure 16 presents the relationships between V/H and magnitude, rupture distance, and Vs₃₀. From Figure 16, the following conclusions can be drawn: (1) V/H increases with magnitude from Mw 5 to Mw 6.5 and decreases from Mw 6.5 to Mw 8 (Figure 16a). (2) Within 20 km of the fault, V/H is broadly distributed; the maximum value (3.43) occurs at a rupture distance of 1.4 km (Figure 16b). (3) For Vs₃₀ < 150 m/s, V/H decreases with increasing Vs₃₀; for Vs₃₀ > 150 m/s, no clear trend is evident (Figure 16c).

Figure 17 presents the relationships between V/H and Fault type: (1) Elevated V/H values are observed for some strike–slip (SS) and reverse-oblique (RV-OBL) events. (2) The average V/H ratios of near-fault ground motions vary among different fault types. Specifically, SS events exhibit the highest average V/H ratio (0.81), followed by RV (0.60), N (0.53), and RV-OBL (0.50) events.

Figure 18 presents V/H distributions for representative Chinese earthquakes: (1) Chi-Chi mainshock within 20 km: V/H > 2/3 in 39% and V/H > 0.8 in 11%; all V/H values are <1.0 (Figure 18a). (2) Wenchuan mainshock within 30 km: V/H > 2/3 in 55% and V/H > 0.9 in 11%; all V/H values are < 1.1 (Figure 18b). (3) Lushan earthquake: all V/H values are <0.7, with 91% < 2/3 (Figure 18c).

From a broader set of records within 100 km and with PGA greater than 0.1 g, the relationship between vertical-to-horizontal PGA ratio (V/H) and rupture distance is shown in Figure 19. Values of V/H exceeding 2/3 are concentrated mainly within 40 km of the fault, with many cases between 1.0 and 1.2; the maximum value (1.6) occurs within approximately 5 km. For conservative design, it is recommended that V/H be taken as not less than 1.0, and for near-fault sites within 20 km, a range between 1.0 and 1.2 is advised.

2.5. Artificial Simulation of Near-Fault Ground Motions

Near-fault pulse-type ground motions exhibit elevated response-spectrum ordinates and thus pose greater potential damage to structures. The 1997 Uniform Building Code (UBC-97) was the first code worldwide to explicitly incorporate near-source effects, introducing near-source factors and clear requirements for the seismic design of structures located close to active faults [65]. In contrast, China’s Code for Seismic Design of Buildings (GB 50011-2010) [66] distinguishes “near” versus “far” events primarily via zonation of design parameters and recommends avoiding construction in near-fault zones; however, it adopts a largely prescriptive (avoidance-based) approach and does not provide detailed design procedures for near-fault conditions. The Specifications for Seismic Design of Highway Bridges (JTG/T B02-01-2008) [67] require that near-fault effects be considered for bridges within 30 km of active faults during seismic safety evaluation, whereas current seismic design codes for highway, municipal, and railway engineering generally do not include explicit near-fault provisions. In particular, the Code for Seismic Design of Railway Engineering does not address bridge design in near-fault regions or related isolation/energy-dissipation strategies, leaving a gap for railway bridges [68].

Given the severity of recent near-fault earthquake damage and the scarcity of recorded near-fault ground motions, there is a pressing need for a robust numerical simulation framework for pulse-type near-fault motions. This study first compares the amplification characteristics of code spectra, far-field spectra, and near-fault spectra to quantify the impact of velocity pulses across structural periods, and then develops nonstationary near-fault simulations matched to target response spectra.

3. Results

3.1. Response-Spectrum Characteristics of Near-Fault Ground Motions

A dataset was compiled from the PEER Strong Motion Database, selecting near-fault double-pulse motions and far-field motions with comparable magnitudes and with peak horizontal acceleration (PHA) ≥ 0.05 g. Near-fault double-pulse records are summarized in Table 2. Faulting mechanisms include strike–slip, reverse, normal–oblique faulting types. Site classes follow the USGS A-D scheme; for analysis, A and B were combined (33 records) and C and D were combined (48 records).

The maximum response-spectrum amplification factor, β_max, depends partly on the economic/design context; China has commonly adopted β_max = 2.25 in recent years [69], while U.S. unified standards often use 2.5 [70]. Table 3 compares β_max for near-fault (pulse and non-pulse), far-field motions, and the code spectrum. The main findings are:

(1). For A+B sites, β_max of near-fault pulse-type motions is lower than that of near-fault non-pulse motions, yet both exceed β_max of far-field motions and of the code spectrum.

(2). For C+D sites, β_max of near-fault pulse-type motions is higher than that of near-fault non-pulse motions, and both exceed β_max of far-field motions and of the code spectrum.

Overall, far-field β_max is close to 2.25, whereas near-fault β_max is appreciably higher. Consequently, the Chinese code value (β_max = 2.25) is suitable for far-field estimates but does not bound near-fault amplification; by contrast, the U.S. practice (β_max = 2.5) generally bounds both.

Figure 20 provides a comparison of amplification factors from three sources: the Chinese code spectrum; the ATC-63 far-field average based on 39 record sets; and near-fault averages derived from 126 pulse-type records grouped by faulting mechanism. The principal observations are:

(1). For periods longer than about 1.5 s, pulse-type motions have amplification factors that exceed both the Chinese code values and the far-field averages, indicating higher seismic demand for long-period structures such as bridges; reliance on the code spectrum alone may be unconservative for near-fault sites.

(2). On average, reverse/normal mechanisms produce the largest amplification beyond 1.5 s; in the long-period range, the mean amplification can exceed the Chinese code value by up to a factor of 3.7, implying the greatest damage potential for this mechanism.

(3). In the long-period range, the ATC-63 far-field average generally exceeds the Chinese code spectrum and roughly envelopes the near-fault averages, except for reverse-faulting cases beyond 1.5 s.

(4). For structures with periods below the site characteristic period, the Chinese code spectrum broadly envelopes the effects across motion types. For long-period structures, the selection of input motions should explicitly account for near-fault effects.

(5). In comparison, the Chinese seismic design code tends to be more conservative in the short-period range, ensuring safety for acceleration-sensitive structures, while the ATC-63 spectrum provides higher amplification in the long-period range, better reflecting displacement-sensitive demands. This difference mainly arises from the distinct design philosophies and ground motion databases used in the two codes—the Chinese code adopts a force-based, safety-oriented approach derived from domestic earthquake records, whereas ATC-63 is performance-based and calibrated using a broader international strong-motion dataset.

3.2. Near-Fault Nonstationary Ground Motion Simulation Matched to Response Spectra

Recorded near-fault motions are scarce, underscoring the need for efficient simulation methods. A common strategy superposes a stochastic high-frequency component on a simple analytical pulse [30]. This approach is limited: single-shape equivalent pulses cannot represent the diversity of observed pulse morphologies, and the high-frequency part must be synthesized from an assumed power-spectral model. Moreover, earthquake ground motion is inherently nonstationary in both amplitude and frequency content, and near-fault records are no exception. Although stationary simulations based on time-invariant power spectra can be generated via trigonometric series, broadly applicable closed-form descriptions of time-varying spectra are lacking. Consequently, accurate simulation of nonstationary near-fault motions remains challenging.

To address this challenge, signal decomposition and reconstruction are employed to generate synthetic ground motions: the original record is decomposed into multiple narrow-band components and then stochastically recombined. Candidate methods include the discrete wavelet transform (DWT), wavelet packet transform (WPT), empirical mode decomposition (EMD), and ensemble EMD (EEMD). EMD/EEMD lack a rigorous mathematical foundation and are sensitive to user-defined parameters, whereas DWT/WPT are not shift-invariant, leading to reconstruction loss and sensitivity to coefficient perturbations [70]. By contrast, the stationary wavelet transform (SWT) is a non-decimated, shift-invariant decomposition that suppresses oscillatory artifacts and reduces reconstruction sensitivity to individual coefficients [71]. These properties motivate a data-driven simulation framework that couples SWT with the Hilbert transform.

3.2.1. Simulation of Near-Fault Nonstationary Ground Motions

(1). Stationary wavelet transform (SWT)

In the Stationary Wavelet Transform (SWT), the choice of the mother wavelet has a significant influence on the effectiveness of signal decomposition. In this study, the Daubechies (db4) wavelet was adopted as the mother wavelet because it provides excellent time–frequency localization characteristics [72]. Based on this, the original signal was decomposed in a dyadic frequency manner into a superposition of several narrow-band (single-component) signals, as shown in Equation (8):

(8) $y_{1} (t) = \sum_{j = 1}^{M} D_{j} (t) + A_{M} (t)$

where

D_{j} (t)

is the detail function at the j level, representing the high-frequency component; M is the number of decomposition levels, which was determined through subsequent experimental analysis;

A_{M} (t)

is the approximation function, representing the low-frequency component, i.e., the trend term.

(2). Hilbert transform.

For any single-component signal c(t), the analytic signal z(t) is defined via the Hilbert transform H(·) [46]:

(9) $z (t) = c (t) + i H [c (t)] = a (t) e^{i ϕ (t)}$

where a(t) and ϕ(t) represent the amplitude and phase varying with time, respectively; the operator H(·) denotes the Hilbert transform, as shown in Equation (10):

(10) $H [c (t)] = \frac{1}{π} P \int_{- \infty}^{\infty} \frac{c (s)}{t - s} d s$

where s is the integration variable; P represents the Cauchy principal value.

According to the analytic form of signal c(t), its instantaneous amplitude and phase can be obtained by Equation (10):

(11) $a (t) = {\sqrt{c (t)^{2} + (H [c (t)])}}^{2}; ϕ (t) = \tan^{- 1} {H [c (t)] / c (t)}$

The derivative of the phase function with respect to time is called the instantaneous frequency, defined as follows:

(12) $ω (t) = \frac{d ϕ (t)}{d t}$

Exploiting SWT for decomposition/reconstruction, the original record is first expressed as a sum of mono-component subseries. Each mono-component is Hilbert-transformed to obtain its instantaneous amplitude $a_{j} (t)$ and frequency $ω_{j} (t)$ , renewed by Equation (13):

(13) $y_{2} (t) = Re [\sum_{j = 1}^{M} a_{j} (t) e^{i \int ω_{j} (t) d t}] + A_{M} (t)$

where Re denotes the real part.

After completing the Hilbert transform and obtaining the instantaneous amplitude and frequency of each component, stochastic parameter calibration was conducted to generate random ground motion samples consistent with the target response spectrum. Specifically, the instantaneous frequency of each component was assumed to follow a Gaussian distribution, with its mean and standard deviation determined from the statistical characteristics of the corresponding component in the original record, while the initial phase angle was assumed to follow a uniform distribution within [0, 2π]. By introducing these stochastic parameters and superposing all single-component signals, nonstationary ground motion samples consistent with the target response spectrum were generated. This process ensures that the artificial ground motions not only preserve the inherent nonstationary characteristics of the original records but also achieve a reasonable reproduction of the target spectral properties. Therefore, the nonstationary ground motion simulation formula based on the SWT and Hilbert transform can be expressed as Equation (14):

(14) $y_{3} (t) = Re [\sum_{j = 1}^{M} a_{j} (t) e^{i [ϕ_{j} (t) + \int {\hat{ω}}_{j} (t) d t]}] + A_{M} (t)$

where

{\hat{ω}}_{j} (t)

is the instantaneous frequency of the j-th order component. Existing studies [50,51,52,73,74] have shown that it is a Gaussian-distributed random variable, and

{\hat{ω}}_{j} (t) ~ N (μ_{j}, σ_{j}^{2})

;

ϕ_{j}

is the uniformly distributed random initial phase angle of the j-th order [0, 2π]; the remaining variables are as described above.

To summarize, simulation of near-fault nonstationary ground motions proceeds in three steps: (1) the original near-fault record is decomposed by the stationary wavelet transform (SWT) into a sum of single-component (narrow-band) subseries; (2) each subseries is subjected to the Hilbert transform to obtain its instantaneous amplitude and phase, from which the instantaneous frequency is computed; (3) stochasticity is introduced by assigning a uniformly distributed initial phase and modeling the instantaneous frequency as Gaussian, thereby generating random realizations of the nonstationary near-fault motion. Notably, this procedure is fully data-driven: decomposition and reconstruction rely solely on features extracted from the original record via the combined SWT-Hilbert analysis.

3.2.2. Spectral Matching of Artificial Near-Fault Motions

Response spectra computed from raw SWT+Hilbert simulations typically deviate from the target spectrum. Spectral matching is achieved by adjusting the amplitude and frequency of each component:

(15) $y_{4} (t) = Re [\sum_{j = 1}^{M} [x_{j} a_{j} (t)] e^{i [ϕ_{j} (t) + x_{n + j} \int {\hat{ω}}_{j} (t) d t]}] + A_{M} (t)$

where x₁, x₂,…, x_n is the adjustment coefficient for the ground motion amplitude component, x_n+₁, x_n+₂,…, x_2n is the adjustment coefficient for the ground motion frequency component, x = { x₁, x₂,…, x_2n}^T is the adjustment coefficient vector, M is the total number of decomposition levels,

a_{j} (t) = {\sqrt{c (t)^{2} + (H [c (t)])}}^{2}

i = \sqrt{- 1}

, and

A_{M} (t)

is the approximation component.

A multivariable optimization theoretical model is then established by constructing adjustment coefficients as follows:

(16) $minimize : V = \sum_{p = 1}^{p} f_{p} (x), f_{p} (x) = \frac{|S [y (x, t), T_{p}] - S_{T} (T_{p})|}{S_{T} (T_{p})} .$

where the constraint condition is

x_{j} > 0, j = 1, 2, \dots, 2 n

S [y (x, t), T_{p}]

is the response spectrum value of the artificial sample at the specified period

T_{p}

S_{T} (T_{p})

is the target response spectrum value at the period

T_{p}

, P is the number of discrete points of the response spectrum, and V is the established objective function.

A nonlinear quasi-Newton algorithm was employed to minimize the objective function V, and the corresponding coefficients were determined to obtain the final artificial ground motions. The workflow is summarized in Figure 21.

4. Discussion

Three recorded near-fault motions (sampling frequency f_s = 100 Hz; duration T = 30 s) are used for validation (Figure 22). Specifically, these three selected ground motions were recorded at seismic stations in Yanshuihe, Bailongjiang and Yangjigou, respectively, based on the site’s seismic safety evaluation [75,76]. All three exhibit pronounced pulse features and non-stationarity.

The three measured samples are decomposed using SWT. Taking wave 1 as an example, the approximation component and the 8 levels of detail components are shown in Figure 23 and Figure 24, respectively. The principle for determining the number of decomposition levels is as follows: since the highest frequency of the approximation component can be determined according to fs/2^j+¹ (j being the number of decomposition levels); when the number of decomposition levels is set to 8, the highest frequency of the approximation component is only 0.19 Hz, which can be used to represent the long-period trend term of the original ground motion record.

Figure 25 shows the instantaneous frequency distribution of the first-order component of wave 1 and its Gaussian fitting result. It can be observed that the instantaneous frequency of this component is approximately Gaussian distributed with slight deviations from the distribution at the tails. To more accurately characterize the mean and variance of the frequency, a truncated Gaussian distribution is used to process the tails, as shown in Figure 26.

Figure 27 compares the artificial and original time histories, and Figure 28 compares the corresponding time–frequency spectrograms. The artificial sample reproduces the temporal evolution of the original record and closely matches its time–frequency spectral characteristics and energy distribution. The same procedure is applied to wave 2 and wave 3; details are omitted for brevity.

Response spectra for the three records are shown in Figure 29. Spectra computed directly from the artificial (pre-adjustment) samples deviate markedly from the target (black curve), particularly at long periods. After applying the optimization introduced in this section, the adjusted spectra align closely with the target (Figure 30).

To quantitatively demonstrate the improvement brought by the adjustment, the root mean square errors (RMSE) between the artificial and target spectra before and after the adjustment were compared, and the results are summarized in Table 4.

The comparison results show a clear reduction in the mean square errors after adjustment for all three waves. Specifically, the accuracy improved by about 40% or more for all three waves, with reductions in error from 0.0812 to 0.0495 for Wave 1, from 0.120 to 0.0671 for Wave 2, and from 0.148 to 0.0568 for Wave 3. These results demonstrate that the adjustment method effectively improves the agreement between the artificial and target spectra and significantly enhances the overall accuracy.

Applying the optimized coefficients from Equations (15) and (16) yields three spectrum-matched nonstationary near-fault samples (Figure 31).

In summary, a data-driven method for simulating nonstationary near-fault ground motions is proposed, leveraging SWT’s shift-invariant decomposition and reconstruction. Component-wise amplitude and frequency adjustments, combined with a multivariable optimization, achieve spectral matching to a prescribed target response spectrum. Validation against recorded motions demonstrates high fidelity and robustness, supporting the method’s use for generating near-fault inputs for bridge seismic analysis.

5. Conclusions

In this paper, records compiled across multiple seismic-station networks are used to quantify near-fault ground motion characteristics and their response-spectrum implications. In particular, relationships between the principal pulse parameters and source-to-site attributes are established, and vertical-component statistics are synthesized. Building on these insights, a data-driven procedure for simulating nonstationary near-fault motions is formulated within an SWT-Hilbert framework, with spectral matching achieved through component-wise amplitude and frequency adjustments and multivariable optimization. The procedure is validated against representative records, demonstrating high fidelity in the time, time–frequency, and spectral domains, leading to the following conclusions:

(1). A baseline-correction procedure combining least-squares and Iwan methods is proposed to mitigate baseline drift in near-fault records, yielding reliable, drift-free time histories.

(2). A statistical analysis was conducted on the long-period velocity pulse characteristics of near-fault ground motion records. The results indicate that only a small proportion of near-fault records exhibit distinct velocity pulses, most of which are unidirectional. Relationships between the key parameters of near-fault velocity pulses—including pulse period (T), peak ground acceleration (PGA), peak ground velocity (PGV), and the PGV/PGA ratio—and earthquake features such as fault distance, magnitude, and shear-wave velocity (Vs₃₀) were established. Moreover, the variation pattern between PGA and the pulse period T was obtained. A method based on Daubechies wavelet decomposition was proposed to extract velocity pulses from ground motion records, providing a foundation for investigating the dynamic response of structures subjected to pulse-like ground motions.

(3). Vertical-component statistics indicate systematic variation of V/H with magnitude, rupture distance, faulting mechanism, and Vs₃₀. Records with V/H > 2/3 occur predominantly within 40 km of the fault; for bridges within 20 km, a conservative input ratio of V/H = 1.0–1.2 is recommended.

(4). Comparison of near-fault amplification (by mechanism), far-field amplification, and code spectra indicates that the Chinese code value β_max = 2.25 is suitable for far-field motions but does not bound near-fault amplification, whereas β_max = 2.5 (U.S. practice) generally provides an envelope for both. Within the SWT-Hilbert decomposition-reconstruction framework, a spectral-matching procedure and artificial-record generation workflow are formulated and validated, demonstrating reliability and practical applicability.

In summary, the proposed method can provide generalized and reliable near-fault ground motions that effectively compensate for the limited availability of recorded earthquake data. The validation results demonstrate that the proposed approach improves the simulation accuracy by about 40% or more, significantly enhancing the agreement between the artificial and target spectra. These artificial motions can serve as valuable inputs for future studies on structural and bridge seismic performance and design. However, the present study is limited by the availability and completeness of near-fault strong-motion records, which restricts the diversity of rupture mechanisms and site conditions represented in the training data. As more near-fault recordings become available in the future, the proposed framework can be further refined and validated to enhance its generalization and applicability.

Author Contributions

Conceptualization: W.H.; methodology: Z.G. and C.L.; validation: W.H.; formal analysis: C.L. and W.W.; resources: B.W.; data curation: Z.G., C.L., W.W., P.S. and Y.Z.; writing—original draft preparation: W.H. and Z.G.; writing—review and editing: W.H., Z.G., C.L., W.W., B.W., P.S. and Y.Z.; project administration: W.H.; funding acquisition: W.H.; Investigation: P.S. and Y.Z. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The original contributions presented in this study are included in this article. Further inquiries can be directed to the corresponding authors.

Acknowledgments

The authors would like to express their gratitude to all those who contributed, directly or indirectly, to this research project.

Conflicts of Interest

Author Yongping Zeng is employed by the company China Railway Eryuan Engineering Group Co., Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Abbreviations

The following abbreviations are used in this manuscript:

SWT	Stationary Wavelet Transform
V/H	Vertical-to-Horizontal
SCEC	Southern California Earthquake Center
FDM	Finite Difference Method
PGA	Peak Ground Acceleration
BAP	Basic Acceleration Processing
LS	Least Squares
PGV	Peak Ground Velocity
CWT	Continuous Wavelet Transform
RV-OBL	Reverse–Oblique
DWT	Discrete Wavelet Transform
WPT	Wavelet Packet Transform
EMD	Empirical Mode Decomposition
EEMD	Ensemble Empirical Mode Decomposition

Footnotes

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Figures and Tables

Figure 1 Schematic of rupture directivity effects.

Figure 2 Orientation of velocity pulses relative to faulting: (a) Strike slip; (b) Dip slip.

Figure 3 Schematic of sensor tilt.

Figure 4 Workflow of the LS-based baseline fit.

Figure 5 Upland earthquake, Rancho Cucamonga-FF station, 90° component: (a) raw acceleration; (b) corrected acceleration; (c) raw velocity; (d) corrected velocity; (e) raw displacement; (f) corrected displacement.

Figure 6 Representative velocity-pulse time history.

Figure 7 Distribution of pulse-type records in the near-fault dataset.

Figure 8 Proportions of unidirectional and bidirectional pulses.

Figure 9 Relationships of rupture distance with different pulse parameters: (a) T; (b) PGA; (c) PGV; (d) PGV/PGA.

Figure 10 Relationships of moment magnitude (Mw) with different pulse parameters: (a) T; (b) PGA; (c) PGV; (d) PGV/PGA.

Figure 11 Relationships of Vs₃₀ with different pulse parameters: (a) T; (b) PGA; (c) PGV; (d) PGV/PGA.

Figure 12 Relationships between PGA and T: (a) all pulses; (b) horizontal pulses; (c) vertical pulses.

Figure 13 Fourth-order Daubechies wavelet.

Figure 14 Extracted horizontal velocity pulses from near-fault records: (a) Wenchuan mainshock record (Mianzhu-Qingping station, 2008); (b) Imperial Valley mainshock record (El Centro station, 1979).

Figure 15 Proportions of V/H in the compiled near-fault dataset.

Figure 16 Relationships of V/H with different pulse parameters: (a) magnitude; (b) rupture distance; (c) faulting mechanism.

Figure 17 Relationships of V/H with fault type.

Figure 18 V/H distributions for representative Chinese earthquakes: (a) Chi-Chi earthquake; (b) Wenchuan earthquake; (c) Lushan earthquake.

Figure 19 Relationship between V/H and rupture distance.

Figure 20 Comparison of amplification factors from different sources.

Figure 21 Flowchart of nonstationary near-fault simulation matched to a target response spectrum.

Figure 22 Three near-fault ground motion records: (a) wave 1; (b) wave 2; (c) wave 3.

Figure 23 Approximation component of the SWT decomposition result.

Figure 24 Detailed component of the WT decomposition result.

Figure 25 Instantaneous frequency distribution of the first-order component and Gaussian fitting.

Figure 26 Truncated instantaneous frequency distribution and Gaussian fitting.

Figure 27 Time histories comparison: (a) original acceleration, (b) artificial acceleration, (c) original velocity, (d) artificial velocity, (e) original displacement, (f) artificial displacement.

Figure 28 Time–frequency comparison: (a) original wave; (b) artificial wave.

Figure 29 Response spectra before adjustment (artificial vs. target).

Figure 30 Response spectra after adjustment (artificial vs. target).

Figure 31 Spectrum-matched nonstationary near-fault samples: (a) wave 1; (b) wave 2; (c) wave 3.

Table 1

Representative near-fault earthquake records.

No.	Earthquake	Year	Date	Station	Magnitude	Fault Type	Rupture Distance (km)	PGA(g)
1	Anza (Horse Canyon)-01	1980	225	Anza—Pinyon Flat	5.19	SS	17.4	0.135
2	Anza (Horse Canyon)-01	1980	225	Anza—Terwilliger Valley	5.19	SS	12.7	0.122
3	Anza-02	2001	1031	Anza—Pinyon Flat	4.92	N-OBL	16.9	0.135
4	Anza-02	2001	1031	Mountain Center—Pine Mtn Rnch	4.92	N-OBL	16.8	0.127
5	Big Bear-01	1992	628	Big Bear Lake—Civic Center	6.46	SS	9.4	0.545
6	Cape Mendocino	1992	425	Fortuna—Fortuna Blvd	7.01	RV	20	0.117
7	Cape Mendocino	1992	425	Rio Dell Overpass—FF	7.01	RV	14.3	0.550
8	Cape Mendocino	1992	425	Petrolia	7.01	RV	8.2	0.661
9	Cape Mendocino	1992	425	Cape Mendocino	7.01	RV	7	1.494
10	Chalfant Valley-01	1986	720	Zack Brothers Ranch	5.77	SS	6.4	0.272

Abbreviations: SS, RV, N, and OBL denote strike–slip, reverse, normal and oblique-slip faulting, respectively.

Table 2

Selected near-fault double-pulse-like ground motions (subset).

No.	PHA/g	PHV/(cm/s)	PHD/cm	M_w	d_r (km)	Site Class	Mechanism
1	0.161	12.4	3.26	6.2	23	A	SS
2	0.132	8.3	1.52	5.7	9.1	A	SS
3	0.434	49.2	7.77	5.7	3.1	B	SS
4	0.314	71.7	25.53	6.5	0.5	D	SS
5	1.039	42	12.39	7.1	8.5	A	RV
6	0.838	166.1	28.78	6.7	7.1	C	RV
7	0.221	28.8	3.95	6	18.3	D	RV
8	0.13	26.3	12.18	6.8	26	D	SS
9	0.476	75.1	22.49	6.1	0.1	C	SS
10	1.285	103.9	23.8	6.7	8	A	RV
11	0.843	129.6	32.68	6.7	6.4	C	RV
12	0.753	84.8	18.68	6.7	8.9	C	RV
13	0.662	89.7	29.55	7.1	9.5	C	RV
14	1.298	80.8	9.63	6.2	0.1	C	SS
15	0.312	39.4	7.66	6.2	3.4	C	SS
16	0.36	76.6	59.02	6.5	4.2	C	SS
17	0.379	90.5	63.03	6.5	1	C	SS
18	0.377	43.9	15.2	6.7	0.7	C	RV
19	0.838	166.1	28.78	6.7	7.1	C	RV
20	0.348	159	184.42	7.6	0.24	C	N-OBL
21	0.486	187.3	266.55	7.6	1.09	C	N-OBL
22	0.333	88.3	86.45	7.6	1.49	C	N-OBL
23	1.01	60	50.15	7.6	1.18	C	N-OBL
24	0.721	97.6	70.31	7.3	1.1	A	SS
25	0.332	61.5	36.4	6.9	13.7	B	RV
26	0.439	109.8	65.89	6.5	1	C	SS
27	0.785	31.9	16.42	7.3	1.1	A	SS
28	0.16	35.9	22.44	6.5	8.5	D	SS
29	0.266	46.8	18.92	6.5	9.3	D	SS
30	1.655	57.5	26.41	6.5	1	D	SS
31	0.41	64.9	27.69	6.5	1	D	SS
32	0.352	71.2	45.8	6.5	21.9	D	SS
33	0.253	48.8	31.54	6.5	7.5	D	SS
34	1.497	127.4	41.01	7.1	8.5	A	RV

Notation: PHA, PHV and PHD denote peak horizontal acceleration, peak horizontal velocity and peak horizontal displacement; M_w is moment magnitude; d_r is rupture distance; site class follows USGS A–D; faulting mechanism: SS = strike–slip, RV = reverse, N-OBL = normal–oblique.

Table 3

Maximum spectral amplification factor βmax.

Site Class	Near-Fault		Far-Field	Code
Site Class	Pulse	Non-Pulse	Non-Pulse	Code
A+B	2.32	2.51	2.20	2.25
C+D	2.41	2.38	2.24	2.25
Average	2.37	2.45	2.22	2.25

Table 4

Comparison of RMSE between artificial and target spectra before and after adjustment.

Ground Motion	Wave 1	Wave 2	Wave 3
Pre-adjustment	0.0812	0.120	0.148
Post-adjustment	0.0495	0.0671	0.0568
Improvement (%)	39.05	44.08	61.62

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Data-Driven Simulation of Near-Fault Ground Motions Using Stationary Wavelet Transform and Hilbert Analysis

Content area

Abstract

Full text

1. Introduction

2. Data and Methods for Near-Fault Ground Motion Analysis

2.1. Near-Fault Ground Motion Characteristics

2.2. Near-Fault Record Selection and Baseline Correction

2.2.1. Requirement for Baseline Correction of Near-Fault Strong-Motion Records

2.2.2. Baseline Correction Methods for Near-Fault Ground Motion Records

2.3. Long-Period Velocity-Pulse Characteristics of Near-Fault Ground Motions

2.4. Vertical-Component Characteristics of Near-Fault Ground Motions

2.5. Artificial Simulation of Near-Fault Ground Motions

3. Results

3.1. Response-Spectrum Characteristics of Near-Fault Ground Motions

3.2. Near-Fault Nonstationary Ground Motion Simulation Matched to Response Spectra

3.2.1. Simulation of Near-Fault Nonstationary Ground Motions

3.2.2. Spectral Matching of Artificial Near-Fault Motions

4. Discussion

5. Conclusions