An Excitation Modification Method for Predicting Subway-Induced Vibrations of Unopened Lines

Abstract

Accurate prediction of subway-induced environmental vibrations for unopened lines remains a significant challenge due to the difficulty in determining appropriate excitation inputs. To address this issue, this study proposes an excitation modification method based on field measurements and numerical simulations. First, field measurements were conducted on a subway line in Shanghai to analyze vibration propagation characteristics and validate a two-dimensional finite element model (FEM). Subsequently, based on the validated model, frequency-band excitation modification formulas were derived. Distinct from existing empirical approaches that often rely on simple statistical scaling, the proposed method utilizes parametric numerical analyses to determine frequency-dependent correction coefficients for four key parameters: tunnel burial depth, tunnel diameter, soil properties, and train speed. The reliability of the proposed method was verified through theoretical analysis and an engineering application. The results demonstrate that the proposed method improves prediction accuracy for tunnels in similar soft soil regions, reducing the prediction error from 10.1% to 5.2% in the engineering case study. Furthermore, parametric sensitivity analysis reveals that ground vibration levels generally decrease with increases in burial depth, tunnel diameter, and soil stiffness, while exhibiting an increase with train speed. This study improves the reliability of vibration prediction in the absence of direct measurements and provides a practical tool for early-stage design and vibration mitigation for unopened lines.

1. Introduction

With the rapid urbanization and development of metro networks worldwide, subway-induced environmental vibrations have become a growing concern [1,2,3]. These vibrations not only affect the foundations of surrounding structures but also disturb residents living near the lines [4,5,6,7]. Therefore, it is essential to develop reliable methods for predicting subway-induced vibrations [8,9,10].

In the past century, empirical approaches were widely used for vibration prediction [11]. However, these methods are highly dependent on the similarity between measured and predicted sites, as well as the accuracy of the underlying data. With the advancement of high-performance computing, numerical methods have emerged as powerful tools for simulating vibration levels [12,13,14,15]. For instance, Galvín and Domínguez [16] demonstrated the reliability of 3D numerical methods for predicting the amplitude of train-induced vibrations. Zhai et al. [17] and El Kacimi et al. [18] developed 3D vehicle–track coupled models to evaluate vibrations generated by moving trains. Tang et al. [19] proposed a 3D time-domain model that accurately captured the primary characteristics of train-induced ground vibrations. Connolly et al. [20] utilized a 3D finite element (FE) model to analyze the effect of embankment stiffness. Yu et al. [12] applied a 3D FEM to study vibration attenuation. The dependence of attenuation behavior on local geology has also been emphasized in measurement-based environmental vibration assessments for other excitation sources, such as blast-induced ground vibrations near critical infrastructure [21]. Kuo K A et al. [22] investigated coupling losses using a 3D building model considering soil–structure interaction. Kouroussis et al. [23] also employed 3D simulations to study vibration wave propagation through track and soil.

However, the high computational cost of 3D models often limits their practical application [24]. To address this, many researchers have turned to 2.5D FEM approaches [25]. Yang and Hung [26] introduced the 2.5D analysis concept to study ground vibrations caused by moving loads. Hung and Yang [27] further discussed key aspects of 2.5D modeling and proposed techniques to improve computational efficiency. Based on the 2.5D analysis concept, Lin et al. [28] developed a method for seismic analysis of underground tunnels. Gao et al. [29] constructed a 2.5D FEM to analyze the vibrations. Some researchers [30,31] explored alternative approaches to optimize the algorithm of the 3D model to improve computational efficiency.

While the 2.5D approach offers a good balance between accuracy and efficiency, the simplicity of 2D models makes them particularly appealing for practical engineering applications, especially during the preliminary design stage. Yang et al. [32] employed a 2D dynamic finite-element analysis to investigate the influence of different parameters on stress paths. Hesami et al. [33] applied a 2D FEM to simulate train-induced vibrations and validated their model against field data. Yang et al. [34] compared 2D and 2.5D FEM results for soil–tunnel systems and concluded that 2D models significantly reduce computational time, although 2.5D models offer greater accuracy. In comparative studies, Vega et al. [35], Xu et al. [30] and Real et al. [36] confirmed that 2D FEMs can provide useful predictions with much lower computational costs than 3D models.

As reviewed above, various numerical approaches have been developed to predict subway-induced vibrations. A key factor for achieving accurate predictions is the definition of excitation input, which is particularly challenging for unopened lines due to the lack of direct operational measurements. Existing studies have employed scaling or analogy concepts or measurement-informed hybrid schemes to transfer or calibrate excitation levels for prediction purposes. However, for unopened metro lines, a practical excitation modification framework that provides explicit frequency-band correction coefficients and separates the contributions of key tunnel–soil–train parameters remain scarce. Moreover, approaches relying on wheel–rail forces or ground-surface measurements may be difficult to apply in early-stage design when such data are unavailable.

To fill in the research gaps, an excitation modification method based on field measurements and finite element modeling is proposed in this study. This approach significantly enhances the prediction accuracy by adjusting reference line data to target line conditions. The paper is organized as follows:

• Field measurement arrangements and vibration characteristic analysis are introduced, followed by the development and validation of a 2D finite element model.
• Mathematical derivation of the frequency-band excitation modification formulas is established, incorporating tunnel burial depth, tunnel diameter, soil properties, and train speed. The method is subsequently verified through theoretical analysis and an engineering case study.
• Parametric sensitivity analyses are conducted to investigate the influence of the four key parameters on ground vibration responses using the proposed method.

2. Field Measurements

2.1. Engineering Background

Field measurements were conducted on a shielded single-track circular subway tunnel in Shanghai. As shown in Figure 1, the tunnel has an outer diameter of 6.2 m and an inner diameter of 5.5 m, with a lining thickness of 350 mm. The track structure consists of a monolithic bed with long sleepers, having a construction height of 750 mm. The burial depth from the ground surface to the rail top is approximately 12 m. Both the track bed and the tunnel lining are constructed using C30 concrete.

2.2. Measurement Arrangement

A DiVimeter-Env-5.0 vibration measurement terminal was employed for data acquisition with a sampling frequency of 250 Hz. Vibration signals were recorded in three orthogonal directions: vertical (Z-axis), transverse horizontal (X-axis, perpendicular to the tunnel axis), and longitudinal horizontal (Y-axis, parallel to the tunnel axis). As illustrated in Figure 2, wireless measurement equipment was installed at specified locations within the tunnel and on the ground surface. Inside the tunnel, measurement points were arranged on both the track bed and the tunnel wall. On the ground surface, measurement points were arranged at horizontal distances of 0 m, 4 m, 8 m, 12 m, 18 m, and 30 m from the tunnel centerline. Specifically, at the 4 m and 8 m distances, sensors were placed on both the east and west sides. In the subsequent analysis, the vibration levels recorded at these symmetrical points were averaged to represent the response at the respective distances. The measuring point at 18 m was a long-term monitoring station.

2.3. Characteristics of Vibrations in the Tunnel

Twenty valid samples with a duration of 15 s each were analyzed. Figure 3 presents the time–history curves of vibration acceleration measured in three directions at the track bed and tunnel wall positions. The results indicate that vibration acceleration levels in the tunnel are approximately 10⁻² m/s². The vertical Z-direction vibration levels are significantly higher than those in the horizontal directions. Therefore, it is reasonable to focus exclusively on Z-directional vibration when evaluating the environmental impact.

To further analyze the spectral characteristics and stability of the source, the 1/3 octave band spectra of vibration levels for the 20 analysis samples are presented in Figure 4. As observed from the spectra, the vibration acceleration levels at both the track bed and tunnel wall are relatively high in the high-frequency range, peaking at approximately 50 Hz, while remaining lower in the low-frequency range below 20 Hz.

A statistical analysis of the Z-direction vibration levels for these 20 samples was conducted to assess data stability, as detailed in

Table 1. Statistics of Z-vibration levels at measurement points.

Measurement Point	Max (dB)	Min (dB)	Mean (dB)	Standard Deviation	Range (dB)
Track Bed	89.0	78.0	83.6	2.7	11.0
Tunnel Wall	77.1	71.4	73.7	1.7	5.6

. The mean Z-vibration level at the track bed is 83.6 dB, which is higher than the 73.7 dB recorded at the tunnel wall. However, regarding data discrete characteristics, the track bed samples exhibit a range of 11.0 dB and a standard deviation of 2.7. In contrast, the tunnel wall samples show a much narrower range of only 5.6 dB and a lower standard deviation of 1.7. These statistical indicators demonstrate that the vibration data collected from the tunnel wall possesses significantly better stability compared to the track bed. Consequently, it is more reasonable to utilize the tunnel wall vibration response as the input excitation for the subsequent numerical analysis.

2.4. Characteristics of Ground Vibrations

To analyze the attenuation characteristics of ground vibrations with increasing distance, peak and effective acceleration measurements at ground-level monitoring points directly above the tunnel centerline were normalized. Figure 5 indicates that vibration amplitudes generally decrease with distance. However, a local vibration amplification phenomenon is observed in the region ranging from 0.5 to 1 times the tunnel burial depth. This amplification at specific frequencies primarily results from the superposition of elastic waves. One portion of the wave propagates directly to this region, while another portion reaches the surface first and then travels horizontally, resulting in their superposition within this region. The shear wave contributes more significantly to this amplification effect compared to the longitudinal wave.

3. Numerical Simulation

3.1. Finite Element Model

3.1.1. Justification for the 2D FEM

To reduce computational complexity, a two-dimensional (2D) finite element method (FEM) model was employed. The three-dimensional tunnel–soil interaction was simplified to a 2D plane-strain problem, as illustrated in Figure 6. While this approach geometrically simplifies the train load as a uniform source acting along the tunnel axis, it does not neglect the complex dynamic interactions of the vehicle–track system. The excitation input applied to the model is derived directly from field measurements of the tunnel wall response. Consequently, the effects of axle spacing, bogie dynamics, and dynamic load amplification are implicitly embedded within the frequency spectrum and amplitude of the measured vibration data. This approach allows for efficient prediction of environmental vibrations while retaining the essential dynamic characteristics of the train load.

Certain physical limitations are inherent to the 2D plane-strain assumption, such as neglecting longitudinal moving load effects and 3D scattering which may lead to potential high-frequency overestimation. However, quantitative comparisons in the literature [30,36] indicate that the discrepancy between 2D and 3D models is generally small and typically falls within the range of 3 to 5 dB in the dominant environmental vibration bands. Furthermore, strictly 2D models tend to yield conservative results in the far field as they neglect geometric attenuation in the longitudinal direction. Therefore, considering the balance between computational efficiency and engineering accuracy, the proposed 2D approach is adopted as a robust and practical tool for predicting subway-induced environmental vibrations.

3.1.2. Model Parameters

The numerical model consisted of an elastic roadbed–tunnel lining–soil system, where the soil was treated as an isotropic material. The influence of groundwater and internal soil pore structures was neglected. The detailed soil properties are provided in

Table 2. The properties of the soil layers.

Layer	Thickness (m)	Depth (m)	Materials	Density (g/cm3)	Shear Wave Velocity (m/s)	Dynamic Young’s Modulus (MPa)	Poisson’s Ratio
①-1	0.7	0.7	Filled soil	1.80	111.8	67.4	0.50
②-1	2.9	3.6	Silty clay	1.86	118.4	78.3	0.50
③	3.9	7.5	Sandy silt	1.93	153.9	137.3	0.50
④-1	10.3	17.8	Mud clay	1.72	158.5	129.2	0.49
⑤-1	8.7	26.5	Clay	1.72	158.5	129.2	0.49
⑥	4	30.5	Yellowish silty clay	1.86	209.7	244.3	0.49
⑦-1	3.5	34	Silty clay	1.86	209.7	244.3	0.49
⑦-2	7.5	41.5	Silty clay with silt	1.95	235.9	323.3	0.49

, and a damping ratio of 0.025 was applied. High Poisson’s ratio values of 0.49 and 0.50 were selected to represent the nearly incompressible behavior of the saturated soft soil layers typical of the Shanghai area. Infinite elements were employed at the bottom and lateral boundaries far from the tunnel centerline, while symmetrical boundary conditions were applied at the boundary near the tunnel axis. The tunnel diameter was set to 6 m, and its burial depth was approximately 12 m. The dimensions of the FEM are set to be 30 m in depth and 50 m in width. The element size is about 0.1 m. The reinforced concrete is modeled as a homogeneous elastic material, without distinguishing concrete and reinforcement.

The FEM of the roadbed–tunnel lining–soil system was developed and divided into networks, as depicted in Figure 7. Measured vertical (Z-direction) acceleration levels on the tunnel wall were used as excitation input to simulate vibration responses at ground-level measurement points.

3.2. Modal Analysis

Modal analysis was conducted to determine natural vibration characteristics and verify the numerical model’s accuracy. The first six vibration modes obtained are presented in Figure 8. The first mode indicates a global translational motion of the soil, confirming that the model dimensions are reasonable. Additionally, good modal coordination between tunnel lining and soil elements suggests high-quality element discretization without any singular or poorly performing elements. The absence of rigid body modes further validates the model’s continuity. The predominant vibration periods obtained from the simulation are consistent with the typical soil vibration characteristics in Shanghai, ranging from 0.35 s to 2.40 s (0.4–3.0 Hz).

3.3. Validation of FEM Simulation

To validate the developed finite element model, a comparison was conducted between the numerically predicted and measured attenuation of vertical ground vibrations. Figure 9 and

Table 3. The comparison of FEM results and field measurement results of Z vibration level at ground measurement points.

Distance (m)	Mean Value of FEM Results (dB)	Mean Value of Field Measurement Results (dB)	Difference (dB)	Error (%)
0	61.6	67.1	−5.5	−8.27
4	60.4	65.3	−4.9	−7.50
8	64.7	66.5	−1.8	−2.66
12	63.6	63.2	0.4	0.58
18	61.4	62.5	−1.1	−1.80
30	61.2	60.8	0.4	0.64

illustrate the results of this comparison. Overall, the numerical predictions show reasonable agreement with the field measurements, with differences within approximately 10% across all monitoring points and reducing to within about 3% at 8 m from the tunnel centerline. The relatively larger deviation at 0 m is attributable to near-field sensitivity, such as the sensor position relative to the tunnel centerline and localized soil heterogeneity. Thus, the developed FEM provides a reasonable basis for predicting ground vibrations and for the subsequent parametric analyses.

4. Excitation Modification Method

4.1. Excitation Modification Formula

The reliability of vibration predictions using FEM simulations depends significantly on the accuracy of excitation input. For subway lines not yet in service, excitation inputs must be estimated by adjusting measured data from similar existing lines. In this study, an excitation modification formula is proposed to improve the accuracy of vibration prediction. The approach analyzes the influence of individual parameters, including burial depth, tunnel diameter, soil properties, and train speed, by altering one variable at a time. Corresponding correction terms are then applied to the frequency-band excitation formulas.

Table 4. Numerical model parameters under different working conditions.

Parameter	Scenario 1 (Different Burial Depths)	Scenario 2 (Different Tunnel Diameters)	Scenario 3 (Different Shear Wave Velocities)
Burial depth	9 m, 18 m, 27 m, 36 m	18 m	18 m
Tunnel diameter	6 m	6 m, 9 m, 12 m, 15 m	6 m
Soil type	Muddy clay	Muddy clay	Silty clay, Muddy clay, Yellowish silty clay, Silty clay with silt
Tunnel section	Circular	Circular	Circular
Track/Lining	C30	C30	C30
Load excitation	1/3 Octave Gaussian Harmonic	1/3 Octave Gaussian Harmonic	1/3 Octave Gaussian Harmonic
Load amplitude	10 kN	10 kN	10 kN
Model size	Horizontal 50 m, Vertical 30 m	Horizontal 50 m, Vertical 30 m	Horizontal 50 m, Vertical 30 m

summarizes the detailed parameter settings under each analysis scenario. The numerical model excitation is calculated according to Equation (1).

VL_Z = VL_Z0 + C

(1)

where VL_Z is the model excitation (dB); VL_Z0 is the measured tunnel wall vibration level of the selected line (dB); and C is the correction term for model excitation, calculated by Equation (2) (dB).

C = C_M + C_D + C_T + C_V

(2)

where C_M, C_D, C_T, and C_V are the correction terms for tunnel burial depth, tunnel diameter, soil shear wave velocity, and train speed, respectively (all in dB).

To determine these correction terms, the relative vibration level, as defined in Equation (3), was calculated for each operating condition relative to a specific baseline.

$$\Delta L=L_1-L_2=20\lg {\displaystyle \frac{a_1}{a_0}}-20\lg {\displaystyle \frac{a_2}{a_0}}=20\lg {\displaystyle \frac{a_1}{a_2}}$$

(3)

where a₁ is the effective acceleration of the varying model, and a₂ is the acceleration of the baseline model.

Using the 9 m burial depth as the baseline, relative vibration levels for models with burial depths of 18 m, 27 m, and 36 m were calculated. Subsequently, linear regression analysis was conducted on these relative levels, as illustrated in Figure 10a. Due to space limitations, only representative frequency band results are presented. The correction coefficients m_ki, which represent the slopes of the linear fitting lines, are summarized in

Table 5. Frequency-band excitation modification coefficients for tunnel burial depth and the corresponding regression statistical indicators.

Frequency (Hz)	mki	R2	95% CI (Lower)	95% CI (Upper)	Frequency (Hz)	mki	R2	95% CI (Lower)	95% CI (Upper)
1	−0.276	0.765	−0.740	0.189	16	−0.286	0.907	−0.565	−0.007
1.25	−0.300	0.731	−0.852	0.253	20	−0.415	0.990	−0.541	−0.289
1.6	−0.342	0.731	−0.973	0.289	25	−0.460	0.965	−0.728	−0.191
2	−0.356	0.949	−0.607	−0.105	32	−0.475	0.952	−0.798	−0.152
2.5	−0.281	0.997	−0.332	−0.231	40	−0.428	0.971	−0.651	−0.204
3.2	−0.186	0.943	−0.325	−0.047	50	−0.201	0.971	−0.306	−0.095
4	−0.304	0.957	−0.500	−0.109	64	−0.219	0.887	−0.456	0.019
5	−0.260	0.986	−0.353	−0.167	80	−0.179	0.923	−0.337	−0.022
6.4	−0.181	0.961	−0.292	−0.069	100	−0.250	0.901	−0.502	0.002
8	−0.149	0.907	−0.293	−0.004	125	−0.324	0.988	−0.434	−0.214
10	−0.222	0.921	−0.421	−0.024	160	−0.217	0.949	−0.370	−0.063
12.5	−0.256	0.855	−0.577	0.064	200	−0.140	0.871	−0.304	0.024

. To evaluate the quality of the regression, statistical indicators including the coefficient of determination R² and the 95% confidence intervals (CI) for the slopes are also reported. As observed in the table, the linear regression models generally exhibit a good fit across most frequency bands.

Based on the linear fitting results, the frequency-band excitation modification formula for burial depth is given by Equation (4).

$$C_{Mi}=m_{ki}·\left(m-m_0\right)$$

(4)

where m is the burial depth of the predicted line; m₀ is the burial depth of the reference line; m_ki is the frequency-band excitation modification coefficient for tunnel burial depth; C_Mi is the excitation modification term for tunnel burial depth.

With the 6 m tunnel diameter serving as the baseline, relative vibration levels for diameters of 9 m, 12 m, and 15 m were calculated. Linear regression analysis was subsequently performed, as illustrated in Figure 10b.

Table 6. Frequency-band excitation modification coefficients for tunnel diameter and the corresponding regression statistical indicators.

Frequency (Hz)	dki	R2	95% CI (Lower)	95% CI (Upper)	Frequency (Hz)	dki	R2	95% CI (Lower)	95% CI (Upper)
1	−0.297	0.996	−0.352	−0.242	16	−0.961	0.884	−2.018	0.097
1.25	−0.281	0.998	−0.322	−0.240	20	−0.987	0.968	−1.534	−0.441
1.6	−0.257	0.993	−0.322	−0.192	25	−1.034	0.978	−1.509	−0.560
2	−0.293	0.996	−0.351	−0.235	32	−1.026	0.991	−1.331	−0.720
2.5	−0.386	0.980	−0.555	−0.216	40	−0.728	0.856	−1.635	0.179
3.2	−0.439	0.899	−0.885	0.008	50	−0.826	0.847	−1.895	0.244
4	−0.542	0.717	−1.578	0.494	64	−1.106	0.997	−1.291	−0.921
5	−0.560	0.809	−1.387	0.268	80	−1.311	1.000	−1.346	−1.275
6.4	−0.631	0.870	−1.373	0.112	100	−1.189	0.980	−1.704	−0.674
8	−0.767	0.851	−1.744	0.210	125	−1.471	0.993	−1.842	−1.100
10	−0.925	0.845	−2.133	0.282	160	−1.447	0.946	−2.497	−0.398
12.5	−0.931	0.862	−2.065	0.203	200	−1.097	0.959	−1.786	−0.407

summarizes the correction coefficients d_ki alongside the corresponding statistical indicators, including the coefficient of determination (R²) and 95% confidence intervals (CI).

Consequently, the frequency-band excitation modification formula is given by Equation (5)

$$C_{Di}=d_{ki}·\left(d-d_0\right)$$

(5)

where d is the tunnel diameter of the predicted line; d₀ is the tunnel diameter of the reference line (analogous measured line); d_ki is the frequency-band correction coefficient for tunnel diameter; C_Di is the excitation modification term for tunnel diameter.

Soil properties are quantified using the Equivalent Shear Wave Velocity V_s, where a lower shear wave velocity indicates softer soil. In Table 4, the shear wave velocities are specified as follows: silty clay at 118.4 m/s, muddy clay at 158.5 m/s, yellowish silty clay at 209.7 m/s, and silty clay with silt at 235.9 m/s. For multi-layered soil strata, the equivalent shear wave velocity is calculated according to Equations (6) and (7).

$$V_s=d_0/t$$

(6)

$$t={\displaystyle \sum _{i=1}^n\left(d_i/V_{si}\right)}$$

(7)

where V_s is equivalent shear wave velocity of the soil (m/s); d₀ is calculation depth, taken as twice the burial depth of the rail top (m); t is shear wave propagation time (s); d_i is thickness of the i-th soil layer (m); V_si is shear wave velocity of the i-th soil layer (m/s); n is number of soil layers.

Using the soil layer with a shear wave velocity of 118.4 m/s serving as the baseline, regression analysis was performed on the relative vibration levels as illustrated in Figure 10c. The slopes of the linear fitting lines for each frequency band are listed in

Table 7. Frequency-band excitation modification coefficients for soil property term and the corresponding regression statistical indicators.

Frequency (Hz)	tki	R2	95% CI (Lower)	95% CI (Upper)	Frequency (Hz)	tki	R2	95% CI (Lower)	95% CI (Upper)
1	−0.105	0.998	−0.119	−0.090	16	−0.020	0.938	−0.035	−0.004
1.25	−0.090	0.987	−0.121	−0.059	20	−0.025	0.998	−0.029	−0.021
1.6	−0.068	0.978	−0.099	−0.037	25	−0.030	0.993	−0.037	−0.022
2	−0.048	0.980	−0.069	−0.027	32	−0.021	0.937	−0.038	−0.004
2.5	−0.050	0.990	−0.065	−0.034	40	0.001	0.029	−0.013	0.015
3.2	−0.048	1.000	−0.051	−0.045	50	0.000	0.002	−0.029	0.028
4	−0.076	0.984	−0.105	−0.046	64	−0.011	0.993	−0.014	−0.008
5	−0.086	0.997	−0.100	−0.072	80	−0.004	0.452	−0.019	0.010
6.4	−0.081	0.991	−0.105	−0.057	100	−0.012	0.999	−0.013	−0.010
8	−0.053	0.967	−0.084	−0.023	125	−0.019	0.984	−0.027	−0.012
10	−0.032	0.957	−0.053	−0.012	160	−0.014	0.988	−0.019	−0.009
12.5	−0.025	0.932	−0.046	−0.005	200	−0.010	0.992	−0.013	−0.007

, together with the coefficient of determination (R²) and 95% confidence intervals (CI).

Consequently, the frequency-band excitation modification formula is given by Equation (8).

$$C_{Ti}=t_{ki}·\left(V_s-V_{s0}\right)$$

(8)

where V_s is the equivalent shear wave velocity of the predicted line; V_s0 is the equivalent shear wave velocity of the reference line; t_ki is the frequency-band correction coefficient for soil properties; C_Ti is the excitation modification term for soil properties.

Using 105 km/h as the baseline speed, relative vibration levels were calculated for various speeds. Linear regression analysis was subsequently performed, as illustrated in Figure 10d. The slopes of the linear fitting lines for each frequency band are listed in

Table 8. Frequency-band excitation modification coefficients for train speed term and the corresponding regression statistical indicators.

Frequency (Hz)	vki	R2	95% CI (Lower)	95% CI (Upper)	Frequency (Hz)	vki	R2	95% CI (Lower)	95% CI (Upper)
1	0.325	0.561	0.090	0.559	16	0.144	0.486	0.023	0.264
1.25	0.069	0.040	−0.205	0.343	20	0.124	0.526	0.028	0.220
1.6	0.247	0.531	0.058	0.437	25	0.052	0.237	−0.024	0.128
2	0.440	0.878	0.307	0.574	32	0.042	0.157	−0.038	0.122
2.5	0.056	0.046	−0.151	0.262	40	0.179	0.575	0.054	0.305
3.2	0.100	0.144	−0.099	0.300	50	0.076	0.225	−0.039	0.192
4	0.076	0.320	−0.014	0.166	64	0.083	0.279	−0.026	0.193
5	0.135	0.153	−0.124	0.394	80	0.055	0.536	0.013	0.096
6.4	0.321	0.740	0.166	0.476	100	0.091	0.782	0.052	0.131
8	0.038	0.042	−0.111	0.187	125	0.054	0.520	0.012	0.096
10	0.046	0.096	−0.069	0.162	160	0.014	0.046	−0.038	0.067
12.5	0.018	0.010	−0.127	0.164	200	0.016	0.052	−0.040	0.072

, accompanied by their respective regression statistics. The lower coefficient of determination R² observed in certain frequency bands reflects the necessary simplification of complex vehicle–track dynamics into a linear model for engineering application. Since train-induced vibrations exhibit oscillatory behavior due to resonance and geometric filtering, a simple linear regression cannot perfectly capture the response in every band. However, this statistical limitation does not compromise the method’s practical validity. As demonstrated in Section 4.2, the aggregate prediction error remains below 2.5%, confirming that band-specific deviations effectively cancel out to yield accurate total vibration levels.

The frequency-band excitation modification formula is given by Equation (9).

$$C_{Vi}=v_{ki}·\left(v-v_0\right)$$

(9)

where $v$ is the train speed of the predicted line; $v_0$ is the train speed of the reference line; $v_{ki}$ is the frequency-band excitation modification coefficient for train speed; C_Vi is the excitation modification term for train speed.

The correction coefficients in Table 5, Table 6, Table 7 and Table 8 exhibit clear frequency dependence. This behavior is because wave propagation and the dynamic impedance of the tunnel–soil system vary with frequency. At higher frequencies, the wavelength becomes shorter and more comparable to geometric dimensions such as tunnel diameter and burial depth. Consequently, changes in geometry and stiffness affect the dynamic stiffness and radiation impedance more strongly, leading to larger correction magnitudes in higher frequency bands. In addition, soil damping and scattering tend to increase with frequency, which further strengthens the frequency dependence of attenuation and transfer from the tunnel to the ground. Using the tunnel diameter term in Table 6 as an example, the increasing magnitude of d_ki with frequency suggests that a larger diameter more effectively suppresses the high-frequency components of tunnel wall vibration, because it increases the effective mass and circumferential stiffness of the coupled system and reduces the efficiency of high-frequency wave radiation into the surrounding soil.

Finally, by substituting the derived correction terms for burial depth C_Mi, tunnel diameter C_Di, soil properties C_Ti, and train speed C_Vi into Equation (2), the total frequency-dependent correction C is determined. Consequently, the precise numerical model excitation for the predicted line can be obtained through Equation (1).

4.2. Verification Through Theoretical Analysis

To validate the proposed formula, theoretical analysis was first performed. Assuming that tunnel wall vibration is primarily governed by the mass of the surrounding soil, the variation rate of vibration response is calculated by Equation (10). The mass involved in the vibration is approximated as the volume of soil extending horizontally by one tunnel diameter and vertically by one burial depth. The resulting variation in tunnel wall vibration level due to the alteration of a single parameter can be estimated using Equation (11).

$$\delta =(1-M_b/M_s) \times 100\%$$

(10)

$$\Delta =20\times \mathrm{l}\mathrm{g}\delta$$

(11)

where $\delta$ is the change in rate of tunnel wall vibration response; M_s and M_b are the mass and the mass change value of the original site involved in vibration, respectively; $\Delta$ is the change value of tunnel wall vibration level induced by 1 m or 1 m/s change in a single parameter change.

As shown in

Table 9. Variation in tunnel wall vibration response induced by a 1 m increase in burial depth.

h0/m	δm	Δ/dB
9	0.89	−1.02
12	0.92	−0.76
15	0.93	−0.60
18	0.94	−0.50
21	0.95	−0.42
24	0.96	−0.37
27	0.96	−0.33
30	0.97	−0.29
33	0.97	−0.27
36	0.97	−0.24
Average	0.95	−0.48

,

Table 10. Variation in tunnel wall vibration response induced by a 1 m increase in tunnel diameter.

d0/m	δ	Δ/dB
6	0.86	−1.34
7	0.88	−1.15
8	0.89	−1.00
9	0.90	−0.89
10	0.91	−0.81
11	0.92	−0.74
12	0.92	−0.68
13	0.93	−0.63
14	0.93	−0.59
15	0.94	−0.56
Average	0.91	−0.84

and

Table 11. Variation in tunnel wall vibration response induced by a 1 m/s increase in shear wave velocity.

Vs0/(m/s)	δv	Δ/dB
120	0.992	−0.073
130	0.992	−0.067
140	0.993	−0.062
150	0.993	−0.058
160	0.994	−0.054
170	0.994	−0.051
180	0.994	−0.048
190	0.995	−0.046
200	0.995	−0.044
210	0.995	−0.041
220	0.995	−0.040
230	0.996	−0.038
240	0.996	−0.036
Average	0.994	−0.051

, theoretical analysis indicates that a 1 m or 1 m/s increase in burial depth, tunnel diameter, and shear wave velocity leads to corresponding reductions in tunnel wall vibration levels of approximately 0.48 dB, 0.84 dB, and 0.051 dB, respectively. In the proposed correction formula, the frequency-band correction coefficients for these three parameters fall within the ranges of 0.15–0.50 dB (burial depth), 0.26–1.47 dB (tunnel diameter), and 0–0.10 dB (shear wave velocity), respectively. These results indicate a strong consistency between the theoretical analysis and the empirical correction model.

The correction term related to train speed is validated using field measurement data. Based on the measured vertical vibration level of the tunnel wall at 105 km/h, the vibration levels at other speeds were predicted using Equation (1). As shown in

Table 12. Comparison of predicted and measured values of Z vibration level of tunnel wall at different speeds.

Speed (km/h)	Measured Z Vibration Level (dB)	Predicted Z Vibration Level (dB)	Difference (dB)	Error
105	84.8	84.8	0.0	0.0%
110	85.0	85.2	0.2	0.2%
115	85.2	85.6	0.4	0.5%
121	86.9	86.1	−0.8	−0.9%
125	86.9	86.3	−0.5	−0.6%
130	87.9	86.7	−1.2	−1.3%
135	85.4	87.0	1.6	1.9%
142	87.3	87.4	0.2	0.2%
153	88.5	88.1	−0.4	−0.5%
165	90.9	88.8	−2.1	−2.4%

, the prediction errors remain within 2.5%, confirming the reliability of the proposed excitation modification formula.

4.3. Verification Through Engineering Examples

A case study from Shanghai Metro Line 9 was used to further verify the excitation correction formula. Field measurements reported by Huang Qiang et al. [37] were used as reference data.

Table 13. Comparison of main parameters of two lines.

Main Parameter	A Section of a Subway Line in Shanghai	A Section of Shanghai Metro Line 9
Tunnel burial depth (m)	12	15
Tunnel diameter (m)	6.2	6.6
Soil equivalent shear wave velocity (m/s)	152.1	151.0
speed (km/h)	40	60
Is it a curve segment?	No	Yes

summarizes the technical parameters of the reference and target lines. Since the studied section includes a curved track, adjustments were made for wheel–rail interaction per standard specifications to account for the curvature effect.

Table 14. Comparison of tunnel wall Z vibration level errors before and after correction (Unit: dB).

Measured Z Vibration Level	Predicted Z Vibration Level Before Correction	Error Before Correction	Predicted Z Vibration Level After Correction	Error After Correction
84.2	75.7	10.1%	79.8	5.2%

presents a comparison of prediction errors before and after applying the correction. Before correction, the error between the value obtained by analogy and the measured result was approximately 10.1%. After applying the proposed excitation modification formula, the error was reduced to 5.2%, demonstrating that the proposed method significantly improves prediction accuracy for the geological and structural conditions typical of Shanghai.

5. Parametric Sensitivity Analysis and Discussion

5.1. Tunnel Burial Depth

The ground Z-vibration levels under different tunnel burial depths were calculated using the proposed excitation modification method. As shown in Figure 11, the ground Z-vibration level generally decreases as the burial depth increases. As the distance from the tunnel centerline increases, the ground vibration induced by metro train operation generally shows an attenuation trend. However, vibration amplification effects can be observed in certain ground areas, and the locations of these amplified vibrations are closely related to the tunnel burial depth, typically occurring at a horizontal distance approximately equal to one time the burial depth.

The influence of different tunnel burial depths on the ground vibration response induced by metro train operation is mainly due to two reasons. First, the increased overburden pressure and soil confinement effectively reduce the vibration response at the source. This phenomenon finds parallels in blast wave dynamics, where confinement and geometric conditions play a decisive role in shaping the vibration response. For instance, Ishchenko et al. [38] analyzed blast wave interactions and highlighted how variable geometric boundaries and confinement significantly alter the wave propagation characteristics. Similarly, in the context of subway tunnels, the enhanced soil confinement at greater depths constrains the vibration energy. Second, the extended propagation path enhances the energy dissipation through geometric spreading and material damping before the waves reach the surface.

To quantify this influence,

Table 15. Correlation between ground z-vibration level and tunnel burial depth (dB/m).

Distance (m)	Depth 9 m~18 m	Depth 18 m~27 m	Depth 27 m~36 m	Mean
0	−0.72	−0.41	−0.27	−0.47
10	−0.95	−0.58	−0.23	−0.59
20	−0.24	−0.54	−0.57	−0.45
30	−0.08	−0.21	−0.37	−0.22
40	−0.18	−0.08	−0.15	−0.14
50	−0.25	−0.12	−0.05	−0.14
60	−0.31	−0.18	−0.05	−0.18
70	−0.35	−0.15	−0.09	−0.20
80	−0.14	−0.41	−0.08	−0.21
Mean	−0.36	−0.30	−0.21	−0.29

examines how the ground Z-vibration level changes with every 1 m increase in burial depth. The analysis leads to two main observations. Burial depth has a stronger effect in the near-field, especially in the regions where vibration amplification occurs. For example, the rate of change at a distance of 10 m reaches −0.59 dB/m, while at a distance of 50 m it decreases to only −0.14 dB/m, which represents about four times the difference. The average reduction rate for shallow tunnels with depths ranging from 9 to 18 m is −0.36 dB/m, whereas for deeper tunnels with depths ranging from 27 to 36 m it decreases to −0.21 dB/m. These results show that increasing burial depth remains an effective approach for reducing vibration.

5.2. Tunnel Diameter

The ground Z-vibration levels under different tunnel diameter conditions were calculated using the proposed excitation modification method. As shown in Figure 12, the ground Z-vibration level generally decreases as the tunnel diameter increases. Similarly to the burial depth effect, vibration attenuation is observed with increasing distance from the tunnel centerline. However, vibration amplification zones are also present, and their locations are closely related to the tunnel diameter. Specifically, for smaller tunnel diameters, the amplification zone tends to occur closer to the tunnel centerline.

The influence of tunnel diameter on ground vibration is primarily attributed to the structural dynamic properties: increasing the tunnel diameter significantly enhances the mass and stiffness of the tunnel structure. Under the same train load, a stiffer and heavier tunnel structure experiences a smaller vibration response, thereby transmitting less energy to the surrounding soil.

To quantify this influence,

Table 16. Correlation between ground z-vibration level and tunnel diameter (dB/m).

Distance (m)	Diameter 6 m~9 m	Diameter 9 m~12 m	Diameter 12 m~15 m	Mean
0	−0.67	−0.39	0.32	−0.25
10	−1.28	−0.69	0.12	−0.61
20	−2.35	−0.58	−0.33	−1.09
30	−1.17	−0.48	−1.15	−0.93
40	−0.79	−0.18	0.09	−0.29
50	−0.77	−0.06	0.34	−0.17
60	−0.85	0.00	0.40	−0.15
70	−0.79	−0.10	0.19	−0.23
80	−0.80	−0.55	0.94	−0.14
Mean	−0.86	−0.23	0.14	−0.32

analyzes the change in ground Z-vibration levels per 1 m increase in diameter. The analysis reveals two distinct patterns. First, the diameter change has a substantial impact within the amplification zones; for instance, at a distance of 20 m, the correlation coefficient reaches −1.09 dB/m, whereas it drops to −0.17 dB/m at 50 m. Second, the sensitivity of vibration levels to diameter changes is much higher for smaller tunnels. The mean correlation for diameters between 6 m and 9 m is −0.86 dB/m, which is significantly stronger than the −0.23 dB/m observed for diameters between 9 m and 12 m. This suggests that increasing the diameter yields the most significant vibration mitigation benefits for smaller tunnels.

5.3. Soil Properties

The ground Z-vibration levels under different soil property conditions (represented by equivalent shear wave velocity, V_s) were calculated using the proposed excitation modification method. As shown in Figure 13, the ground Z-vibration level decreases as the shear wave velocity increases (i.e., as the soil becomes stiffer). Unlike burial depth and tunnel diameter, the shear wave velocity appears to have little correlation with the location of amplification zones or the rate of attenuation with distance. The attenuation trends and peak locations remain consistent across different soil stiffness conditions.

The influence of different soil properties on the ground vibration response induced by metro train operation is mainly due to two reasons: first, the higher density and elastic modulus of stiffer soils (higher shear wave velocity) increase the acoustic impedance of the system, effectively reducing the vibration response at the source; second, softer soils are less effective at filtering low-frequency vibration components, resulting in higher residual vibration energy at the ground surface compared to stiffer soils.

To quantify this influence,

Table 17. Correlation between ground z-vibration level and shear wave velocity (10⁻¹ dB/(m·s⁻¹)).

Distance (m)	Vs 118.4~158.5 m/s	Vs 158.5~209.7 m/s	Vs 209.7~235.9 m/s	Mean
0	−0.47	−0.57	−0.48	−0.51
10	−0.41	−0.38	−0.33	−0.37
20	−0.40	−0.47	−0.47	−0.45
30	−0.29	−0.35	−0.38	−0.34
40	−0.26	−0.30	−0.33	−0.30
50	−0.23	−0.32	−0.36	−0.30
60	−0.26	−0.34	−0.36	−0.32
70	−0.19	−0.28	−0.34	−0.27
80	−0.14	−0.29	−0.43	−0.29
Mean	−0.29	−0.37	−0.39	−0.35

analyzes the change in ground Z-vibration levels per 10 m/s increase in shear wave velocity. The analysis yields two key findings. Soil stiffness has a dominant effect in the near-field. For instance, the correlation coefficient is −0.51 dB/(10 m/s) at a distance of 0 m, compared to −0.29 dB/(10 m/s) at 80 m. Furthermore, the sensitivity of vibration levels increases as the soil becomes stiffer. The mean reduction rate for softer soils (118–158 m/s) is −0.29 dB/(10 m/s), whereas it increases to −0.39 dB/(10 m/s) for stiffer soils (209–235 m/s). This indicates that improving soil stiffness (e.g., via ground treatment) is an effective measure for vibration mitigation, particularly in the near-field.

5.4. Train Speed

Train speed significantly influences the acceleration vibration response of both the track bed and the tunnel wall. As shown in

Table 18. Peak acceleration of measurement points at different train speeds.

Speed (km/h)	Peak Acceleration of Measurement Points (m/s2)
	Track Bed Z-Direction	Track Bed X-Direction	Tunnel Wall Z-Direction	Tunnel Wall X-Direction
105	1.35	1.56	0.22	0.11
110	1.22	1.74	0.24	0.10
115	1.46	1.63	0.25	0.11
121	1.35	1.70	0.25	0.14
125	1.29	1.87	0.23	0.14
130	1.58	1.58	0.26	0.16
135	1.54	1.69	0.24	0.10
142	1.69	1.91	0.28	0.15
153	1.80	2.60	0.29	0.22
165	1.94	2.24	0.30	0.15

, as the train speed increases, the peak accelerations in both the vertical Z and horizontal X directions for the track bed and tunnel wall measurement points exhibit a gradual increase, although oscillatory fluctuations are observed under certain speed conditions. This phenomenon is primarily attributed to the intensified impact between the wheels and rails caused by the faster movement of the load at higher speeds, which generates a stronger vibration response.

The vibration magnitude at the track bed is consistently higher than that at the tunnel wall. Specifically, the horizontal X vibration exceeds the vertical Z vibration at the track bed, whereas the opposite is observed at the tunnel wall where the Z-vibration dominates. This is because the train generates significant horizontal impact on the track within the analyzed section, inducing large horizontal vibrations. However, the attenuation of horizontal vibration during transmission from the track bed to the tunnel wall is greater than that of vertical vibration.

The Z-vibration level generally exhibits a linear increasing relationship with train speed, as illustrated in Figure 14. However, oscillatory fluctuations are evident in specific speed ranges, such as speeds above 130 km/h for the track bed and between 130 km/h and 140 km/h for the tunnel wall. The vibration response is minimal at 105 km/h for both locations. For the track bed, the maximum response occurs at 130 km/h, showing an increase of approximately 6 dB compared to the 105 km/h case. For the tunnel wall, the maximum response is observed at 165 km/h, also representing an increase of approximately 6 dB relative to the 105 km/h baseline.

6. Discussion

The proposed excitation modification method significantly enhances prediction accuracy for unopened subway lines, reducing the error from 10.1% to 5.2% in the engineering case study. The limitations and scope of the methodology are discussed below.

6.1. Assumptions of Linearity and Superposition

First, the framework relies on linear regression and linear superposition. This implies an approximately linear relationship between vibration response and the key parameters, namely burial depth, diameter, soil properties, and train speed, assuming these parameters act independently. While theoretical considerations and field validation support this approximation within the investigated ranges, such as train speeds of 105 to 165 km/h, this approach simplifies complex physical interactions. It explicitly neglects potential coupling effects, such as the interaction between burial depth and soil stiffness or between train speed and track-tunnel resonance. Consequently, prediction accuracy may decrease if parameter variations are substantial or if pronounced nonlinear parameter interactions occur. Future research could refine this method by adopting piecewise linear fitting or introducing coupling coefficients to account for these complexities.

6.2. Physical Modeling Simplifications

The soil was modeled as a linear elastic medium. Although subway-induced environmental vibrations typically generate small-strain shear in the far-field, thereby justifying the linear assumption, soil may exhibit nonlinear behavior, such as stiffness degradation and increased damping, in the immediate vicinity of the tunnel or in extremely soft layers. Explicitly modeling such nonlinearities would likely result in higher energy dissipation and more rapid attenuation. Consequently, the correction coefficients derived under the linear assumption tend to be slightly conservative, effectively providing a safe, upper-bound estimation for environmental impact assessment.

6.3. Scope of Applicability and Generalization

The specific correction coefficients derived in this study are based on field measurements and geological conditions in Shanghai, which are characterized by saturated soft clay. Therefore, these specific values are most applicable to subway lines in regions with similar geological features. For significantly different conditions, such as hard rock or dry sandy soil, the specific coefficients may not be directly applicable due to differences in stiffness, damping, and wave propagation rates. However, the methodology itself remains generalizable. Engineers working in diverse geological environments can utilize the proposed procedural framework by establishing a baseline via local measurements and conducting parametric FEM analysis to derive site-specific correction coefficients tailored to their local conditions.

7. Conclusions

To address the challenge of determining accurate excitation inputs for unopened subway lines, this paper proposes an excitation modification method based on field measurements and finite element modeling. The study validates a 2D numerical model against field data and derives frequency-band correction equations for tunnel burial depth, tunnel diameter, soil properties, and train speed. An engineering application on Shanghai Metro Line 9 is used to verify the effectiveness of this method, leading to the following key conclusions:

(1)

Field measurements indicate that vertical vibration dominates the tunnel response, and the vibration at the tunnel wall is more stable than at the track bed. The established 2D FEM was validated against field data, showing high accuracy with a prediction error of less than 3% in the near-field region, proving its suitability for parametric analysis.

(2)

The proposed excitation modification method improves prediction accuracy. Application to the engineering case study demonstrates that, compared to the traditional analogy method, the prediction error is reduced from 10.1% to 5.2%. This confirms that incorporating correction terms for key parameters effectively aligns the reference line data with the target line conditions, particularly for tunnels situated in soft soil environments similar to the verified case.

(3)

Ground vibration levels generally decrease with increasing tunnel burial depth, tunnel diameter, and soil shear wave velocity, while increasing linearly with train speed. Distinct vibration amplification zones are observed at specific distances correlated with burial depth and diameter. Furthermore, increasing the tunnel diameter yields the most significant mitigation benefits for smaller tunnels (6 m–9 m), and improving soil stiffness is particularly dominant in reducing near-field vibrations.

In conclusion, the proposed excitation modification method successfully addresses the inaccuracies in vibration prediction caused by improper excitation inputs and holds great appeal for engineering applications. By enabling the precise adjustment of measured data from existing lines, this method provides an efficient tool for the early-stage environmental impact assessment of future subway developments without requiring computationally expensive 3D models.