Ground-Borne Vibrations Induced by Railway Traffic: Impact, Prediction, Mitigation and Future Perspectives

Abstract

Ground-borne vibrations caused by railway traffic represent a significant environmental concern, particularly in densely populated or vibration-sensitive urban areas. These phenomena can lead to discomfort and annoyance among residents, interfere with the operation of sensitive equipment, and even threaten the integrity of heritage sites or structurally vulnerable buildings and infrastructures. Building on these concerns, this paper presents a comprehensive review of the current state of knowledge on the subject. It begins by examining the impacts of ground-borne vibrations on both people and structures, followed by an overview of the regulatory frameworks implemented in different countries to manage these effects, with a focus on four examples from Europe and North America. The review then systematically explores the key factors associated with the generation and propagation of ground-borne noise and vibrations. Furthermore, prediction methodologies are categorised into four groups—analytical and semi-analytical, numerical, empirical and AI-based models—and critically assessed. Finally, the paper reviews mitigation strategies applied at the source, along the propagation path, and at the receiver, assessing their effectiveness in reducing the identified impacts.

1. Introduction

1.1. General Overview: Problem Description

The numerous economic, social, and environmental benefits of urban rail transport make it a widely recognised and highly efficient mode of transportation, particularly in densely populated urban areas. However, the exploration of rail infrastructures is associated with the ground-borne noise and vibrations generated by passing rolling stock, which propagate through the ground and reach surrounding buildings. This transmitted energy can be perceived inside the built environment as structural vibrations and/or re-radiated noise. These phenomena may lead to discomfort among occupants, interfere with the operation of vibration-sensitive equipment, and, under extreme conditions, cause architectural damage.

From a general point of view, the problem of ground-borne vibration generated by railway traffic can be summarised as follows: (i) the movement of the train along the track generates a source of vibration; (ii) this vibration propagates through the surrounding ground; and (iii) the resulting vibration field reaches the surrounding buildings, inducing structural vibrations and re-radiated noise, along with all the associated drawbacks previously mentioned. A schematic representation of the problem is illustrated in Figure 1.

1.2. Impact on Residents and Structures

Light rail traffic can induce vibration velocity levels higher than 76 dB in the areas surrounding the track [1], as can be seen in Figure 2. This value is higher than the human perception threshold, which is approximately 70 dB (ref. ${10}^{-8}$ m/s) [1]. However, when dealing with freight traffic, vibration levels can increase up to 93 dB (ref. ${10}^{-8}$ m/s), a high range value when compared to human perception [1].

In terms of long-term effects, vibrations generated by railway traffic generally do not pose a threat to the structural integrity of buildings. They mainly constitute an environmental disturbance, with resident discomfort and the malfunction of sensitive equipment being the primary concerns. Damage to buildings would require extremely high vibration levels, which occur only under unusual operating conditions, as indicated by the typical ground-borne vibration levels in Figure 2.

Although residents often worry about potential structural damage, such fears are generally unfounded, as vibration levels capable of causing damage are one to two orders of magnitude higher than those associated with human perception [2,3]. Therefore, structural damage from railway-induced vibrations is not expected. Nevertheless, historic and heritage structures require special consideration due to their age, materials, and construction techniques, which can make them more sensitive to long-term cumulative vibration effects [4].

As depicted in Figure 2, vibrations caused by railway traffic are usually well below the threshold for structural damage in buildings, making them primarily a source of discomfort rather than a structural concern. However, persistent vibrations may cause material fatigue, potentially leading to the development of cracks and long-term structural weaknesses [5]. Several issues are associated with structural vibrations, including discomfort and disturbance in daily activities, which typically reduce the life quality and work performance. In extreme cases, prolonged exposure can even lead to serious health problems [6,7].

The human response to vibrations is influenced by various factors. Some of these are related to the characteristics of the vibration itself, which can be objectively measured, such as amplitude, duration and frequency content. On the other hand, subjective factors (from economic, personal or social nature) lead to a variable perception of vibrations from individual to individual [8]. To overcome this particularity, a statistical approach based on exposed–response curves’ cause–effect can be an effective solution [9,10].

The study of exposed–response relationships involves the consideration and analysis of different levels of discomfort revealed by the building’s users when subjected to prescribed levels of structural vibration, quantifying the percentage of individuals who perceive the vibrations. Recent studies, which have contributed to establishing vibration velocity limit values in international manuals, such as Federal Transit Administration (FTA) [11] and Federal Railroad Administration (FRA) [12], indicated that floors with vibration levels higher than 83 dB (dB ref. $1\times {10}^{-8}$ m/s) are intolerable from a comfort perspective, particularly for frequent events.

Table 1. Vibration and ground-borne noise levels and associated human responses [11,12].

Vibration Velocity Level (Running RMS) (ref. ${10}^{-8}$ m/s)	Ground-Borne Noise Level		Human Response
	Low Freq. * (dBA)	Mid Freq. ** (dBA)
73.1 VdB	25	40	Approximate threshold of vibration perception for many humans. Low-frequency sound usually inaudible, mid-frequency sound excessive for quiet sleeping areas.
83.1 VdB	35	50	Approximate dividing line between barely perceptible and distinctly perceptible. Many people find train vibration at this level unacceptable. Low-frequency noise acceptable for sleeping areas, mid-frequency noise annoying in most quiet occupied areas.
93.1 VdB	45	60	Vibration acceptable only if there are an infrequent number of events per day. Low-frequency noise unacceptable for sleeping areas, mid-frequency noise unacceptable, even for infrequent events with institutional land uses such as schools and churches.

* Approximate noise level when vibration spectrum peak is near 30 Hz. ** Approximate noise level when vibration spectrum peak is near 60 Hz.

summarises the human response to different levels of vibrations and ground-borne noise induced by railway traffic. It is important to note that noise values only represent the noise levels directly generated by the vibration of the walls and slabs that delimit dwellings and do not account for other sources of noise such as direct noise induced by traffic.

The information in Table 1 indicates that the impact of vibration and ground-borne noise is frequency dependent, as human response varies accordingly. For example, vibration-induced noise from structural components may cause annoyance even when the vibration itself is imperceptible, whereas low-frequency vibrations can be disturbing even if the associated ground-borne noise is minimal. The values presented in Table 1 refer to residential buildings. Naturally, buildings with different functions have distinct tolerance levels for vibrations. For a frequency range of up to 200 Hz, railway-induced vibrations can interfere with the normal operation of sensitive equipment inside buildings near the infrastructure. These vibrations may damage equipment or hinder their proper functioning. Possible effects include reduced precision and reading speed, or even an inability to take readings, thus rendering the equipment inoperable. Therefore, when it comes to certain more sensitive equipment, vibrations can make their use impractical [13].

For precision instruments, such as those used in research laboratories, microelectronics, nanotechnology, medical, and biopharmaceutical industries, including electron microscopes, even vibration levels well below the threshold of human perception can pose a serious concern. In Figure 3, the images captured by an electron microscope with a magnification of 150,000 are shown, under normal conditions and when disturbance occurs in its operation due to vibration [13].

2. Regulatory Framework

2.1. Generalities

The phenomena associated with continued exposure to ground-borne vibrations induced by railway traffic are addressed within a regulatory framework, which is closely linked to the frequency range in which these vibrations occur. Specifically, building vibrations are perceived by the human body as mechanical vibrations within the frequency range of 1 Hz to 80 Hz, whereas in the range of 16 Hz to 250 Hz, such vibrations may be perceived as sound (ground-borne noise), induced by the vibration of walls, floors, and ceilings [7]. The following sections provide a general overview of the main regulations adopted in different countries. Nevertheless, consultation of the RIVAS project deliverables [14] is recommended in order to access a more exhaustive compilation and description of standards, regulations, and guidelines. An initial systematisation of the key features of the different regulations is presented later. It is important to highlight that there is a wide variety of descriptors (e.g., maximum running RMS, vibration dose value) based on different physical quantities (such as acceleration or vibration velocity). These differences stem from the historical development of standards, the dominant application areas in each country, and the frequency ranges considered relevant.

2.2. U.S. Regulations

In the United States, FTA manual [11] serves as a comprehensive guideline for transport infrastructure projects. It provides recommendations and analysis procedures for addressing ground-borne noise and vibrations induced by rail traffic that follow the actual state of the art.

The FTA manual classifies the analysis into three levels: a screening procedure, general vibration assessment, and detailed vibration analysis. The screening procedure serves as the initial step to delineate the study area for any subsequent vibration impact evaluation. The general vibration assessment and detailed vibration analysis are used to determine the extent and severity of the impact. In most cases, the general vibration assessment is sufficient. However, if the project is located near vibration-sensitive land uses, and the general assessment suggests that the impact may be significant, it is advisable to conduct a more comprehensive, detailed analysis to accurately characterise the potential effects.

Depending on the level of analysis, different vibration criteria are established. For the general vibration assessment, the FTA manual outlines distinct thresholds for ground-borne noise and vibrations, taking into account factors such as land use and the event frequency (e.g., train passages). These limit values are detailed in

Table 2. Impact criteria for general vibration assessment of Indoor Ground-Borne Vibration (GBV) and Ground-Borne Noise (GBN) [11].

Land Use Category	GBV Impact Levels (VdB) (${\mathit{dB}}_{\mathit{ref}}={10}^{-8}$ m/s)			GBN Impact Levels (dBA) (${\mathit{dB}}_{\mathit{ref}}=20\times {10}^{-6}$ Pa)
	Frequent Events 1	Occasional Events 2	Infrequent Events 3	Frequent Events 1	Occasional Events 2	Infrequent Events 3
Category I: Buildings where vibration would interfere with interior operations.	73.1 VdB	73.1 VdB	73.1 VdB	N/A	N/A	N/A
Category II: Residences and buildings where people normally sleep.	80.1 VdB	83.1 VdB	88.1 VdB	35 dBA	38 dBA	43 dBA
Category III: Institutional land uses with primarily daytime use.	83.1 VdB	86.1 VdB	91.1 VdB	40 dBA	43 dBA	48 dBA

¹ Frequent Events: more than 70 vibration events of the same kind per day. ² Occasional Events: between 30 and 70 vibration events of the same kind per day. ³ Infrequent Events: fewer than 30 vibration events of the same kind per day.

.

Ground-borne vibration limits are expressed in terms of Running RMS (Root Mean Square) of the vibration velocity (in dB, ref. ${10}^{-8}$ m/s). A one-second averaging period is commonly used to calculate the mean squared amplitude of the vibration signal. Ground-borne noise limits are expressed as A-weighting sound pressure in dB units.

With regard to ground-borne noise, the equivalent pressure level spectra should be assessed against the threshold values defined in Table 2.

2.3. Portugal

In Portugal, there is currently no specific regulatory framework addressing ground-borne noise and vibrations induced by railway traffic. However, the National Laboratory of Civil Engineering (LNEC) provides a set of guidelines aimed at ensuring human comfort in residential and intellectual environments during the exploration of a railway infrastructure [15].

LNEC distinguishes between underground and at-grade railway projects, proposing different threshold limits and evaluation criteria for ground-borne noise and vibrations. The first set of criteria is summarised in

Table 3. Ground-borne noise and vibrations criteria proposed by LNEC [15].

Track Type	Ground-Borne Vibration		Ground-Borne Noise
	Day	Night	Indirect Criteria	Direct Criteria
Underground	0.28 mm/s		0.025 mm/s	27 dB(A)
At-grade	0.28 mm/s	0.11 mm/s	0.05 mm/s	36 dB(A)

.

Thresholds for ground-borne vibrations are defined in terms of the average root mean square (RMS) vibration velocity, expressed in millimetres per second (mm/s). These values apply to the vertical or horizontal component of the velocity, whichever is more significant.

• Indirectly, by limiting the RMS vibration velocity levels integrated over 1/3-octave frequency bands with centre frequencies between 16 Hz and 250 Hz, measured at the building’s structural slabs.
• Directly, through acoustic measurements, by calculating the equivalent continuous A-weighted sound pressure level (dB(A)) for the same 1/3-octave bands.

The second set of criteria applies to frequency-domain analyses and is valid for both underground and at-grade railway lines. LNEC recommends the use of the base curve from ISO 2631-2: 1989 [8], with multiplicative factors to account for intermittent vibrations in residential buildings. The allowable limits depend on the time of day and the centre frequency of each 1/3-octave band, as detailed below:

• Night period (20:00–08:00):

  -

  0.14 mm/s for centre frequencies between 8 Hz and 80 Hz

  -

  0.40 mm/s at 2 Hz

  -

  0.80 mm/s at 1 Hz

• Daytime period (08:00–20:00):

  -

  0.20 mm/s for centre frequencies between 8 Hz and 80 Hz

  -

  0.56 mm/s at 2 Hz

  -

  1.12 mm/s at 1 Hz

To assess compliance with the various criteria presented, the average value of the 10 most unfavourable vehicle passages, in both directions, shall be determined. For each individual passage, the measurement time must correspond to at least a 10 dB decrease from the maximum level [15].

2.4. Germany

At present, Germany does not have specific regulations for the assessment of railway-induced ground-borne vibrations. Nevertheless, standard DIN 4150-2 [16] is commonly employed to evaluate the effects of vibrations on people within buildings and on structures, respectively.

DIN 4150-2 provides guidelines for assessing vibrations in the frequency range of 1 Hz to 80 Hz that affect occupants of buildings. This standard recommends the use of the running RMS value of the frequency-weighted vibration signal, defined by the following expression [16]:

$$K{B}_F(t)=\sqrt{{\displaystyle \frac{1}{\tau }}{\int }_0^{t}K{B}^2(\xi )e^{-{\displaystyle \frac{t-\xi }{\tau }}}d\xi },$$

(1)

where the symbol $K{B}_F(t)$ denotes the weighted vibration velocity, calculated using a time constant $\tau =0.125 \mathrm{s}$. It should be noted that the weighted velocity signal $KB(t)$ is derived by frequency (f) weighting (filtering) as:

$$H_{KB}(f)={\displaystyle \frac{1}{\sqrt{1+{\left(\frac{5.6}{f}\right)}^2}}}$$

(2)

Although no unit is explicitly stated in the standards, the associated unit is clearly $\mathrm{m}/\mathrm{s}$ (more commonly expressed as $\mathrm{mm}/\mathrm{s}$). The level of comfort can then be assessed by comparing the maximum value ${K{B}_F}_{\max }$ with three guideline limits denoted by $A_u$, $A_o$, and $A_r$, which are used both for an overall evaluation and for short-term frequency vibrations, as shown in

Table 4. Reference values “A” for the assessment of vibration emissions in dwellings and similarly used rooms—adapted from [16].

Line	Location of Effect	Day			Night
		${\mathit{A}}_{\mathit{u}}$	${\mathit{A}}_{\mathit{o}}$	${\mathit{A}}_{\mathit{r}}$	${\mathit{A}}_{\mathit{u}}$	${\mathit{A}}_{\mathit{o}}$	${\mathit{A}}_{\mathit{r}}$
1	Locations of impact, in whose surroundings only commercial facilities and, if applicable, exceptionally residences for owners and managers of the businesses as well as for supervisory and on-call personnel are accommodated.	0.4	6	0.2	0.3	0.6	0.15
2	Locations of impact, in whose surroundings predominantly commercial facilities are accommodated.	0.3	6	0.15	0.2	0.4	0.1
3	Locations of impact, in whose surroundings neither predominantly commercial facilities nor predominantly residences are accommodated.	0.2	5	0.1	0.15	0.3	0.07
4	Locations of impact, in whose surroundings predominantly or exclusively residences are accommodated.	0.15	3	0.07	0.1	0.2	0.05
5	Particularly vulnerable locations of impact, e.g., in hospitals, health clinics, as far as they are located in specially designated special areas.	0.1	3	0.05	0.1	0.15	0.05

. It should be noted that $A_r$, $A_u$, and $A_o$ correspond to the lower reference value, upper reference value, and the reference value for rarely occurring impacts, respectively.

2.5. UK

The BS 6472 [17] defines the term vibration dose value (VDV) as a metric that provides a consistent assessment of continuous, intermittent, occasional, and impulsive vibrations, and correlates well with subjective human response. The VDV is given by Equation (3):

$$\begin{array}{c}\hfill {\mathrm{VDV}}_{\mathrm{b}/\mathrm{d}, \mathrm{day}/\mathrm{night}}={\left({\int }_0^T{a}^4(t)dt\right)}^{0.25},\end{array}$$

(3)

where $a(t)$ is the frequency-weighted acceleration, and T is the total period during the day or night over which vibration may occur.

According to BS 6472 [17], the VDV associated with a perception of “uncomfortable” falls within the range of 0.8 to 1.6 m·s^−1.75. The assessment is based on the probability that the experienced vibration dose will provoke adverse comment from those affected, as summarised in

Table 5. Vibration dose value ranges which might result in various probabilities of adverse comment within residential buildings [17].

Place and Time	Low Probability of Adverse Comment m·s−1.75	Adverse Comment Possible m·s−1.75	Adverse Comment Probable m·s−1.75
Residential buildings 16 h day	0.2 to 0.4	0.4 to 0.8	0.8 to 1.6
Residential buildings 8 h night	0.1 to 0.2	0.2 to 0.4	0.4 to 0.8

. Below the threshold of 0.2 m·s^−1.75, annoyance is not expected, whereas above 1.6 m·s^−1.75, it is considered very likely that residents will be disturbed by the perceived vibrations.

2.6. Summary

This section presents a summarised comparison of the standards provided by different countries. As shown in

Table 6. Standards and guidelines (adapted from RIVAS D1.4 [14]).

Key Feature	USA FTA (2018)	Portugal LNEC (2024)	Germany DIN 4150-2 (1999)	UK BS 6472-1 (2008)
Frequency range	1–80 Hz	1–80 Hz	1–80 Hz	0.5–80 Hz
Measured quantity	Velocity	Velocity	Velocity	Acceleration
Indicator	General assessment: maximum running RMS	Average root mean square (RMS) vibration velocity	Maximum weight vibration velocity	Vibration dose value
Measurement	Near the centre of a floor span where the vibration amplitude is the highest	Vertical or horizontal component of the velocity (whichever is more significant)	Three directions with horizontal x- and y-axes parallel to the walls as much as possible	Highest expected level. Central part of the floor (one or two measurements)

, key features including frequency range, measured quantity, indicator, and measurement are evaluated across the considered national standards.

3. Generation and Propagation of Ground-Borne Noise and Vibrations

3.1. Sources of Railway-Induced Ground-Borne Vibration

Ground-borne vibrations are generated through the dynamic interaction between the train and the track. This dynamic action encompasses multiple components, each linked to specific excitation mechanisms, which can be classified into three types: quasi-static, dynamic and parametric excitations [18].

In general terms, quasi-static excitation results from the movement of loads corresponding to the self-weight of the train, distributed over its axles. This type of excitation exhibits a constant magnitude over time, and the dynamic character of the system’s response arises from the temporal variation in stress and strain fields at a fixed point in the domain as the vehicle moves. Quasi-static excitation typically occurs at low frequencies (below 20 Hz) and is mainly governed by axle loads and track stiffness.

In contrast to quasi-static excitation, dynamic excitation is caused by the inertial forces generated within the vehicle, as illustrated in Figure 4. This type of excitation can stem from various sources, the most common being geometric or stiffness irregularities in the track, as well as imperfections in the train wheels. Dynamic excitation spans a wide frequency range, from low to high frequencies, with the dominant frequencies depending on the specific source, train speed, train type, among others. For example, it has been suggested that wheel unevenness plays a more significant role than track irregularities in the case of freight traffic [18].

Among the mechanisms of dynamic excitation, parametric excitation is often referenced. It originates from the discrete nature of rail support (except in cases where embedded rail systems are used). This condition gives rise to a dynamic interaction force that can be distinguished from the other types due to its dependence on a geometric parameter of the track. As an illustrative example, the sleeper-passing frequency can be simply estimated as the ratio ${\displaystyle \frac{\nu }{d}}$, where d is the sleeper spacing and $\nu$ the train speed [18]. Nevertheless, only residual significance is attributed to the latter component in the presence of irregularities in the track and the vehicle [19], which aligns with the classification of excitation mechanisms described by Auersch [20].

From the previously outlined concepts, it is evident that the condition of both the train and the track significantly influences the generation of ground-borne vibrations. These conditions affect the characteristics and intensity of the dynamic interactions at the wheel-rail interface. In this context,

Table 7. Source-related factors influencing vibration levels.

Component	Factor	Influence	References
Rolling stock	Speed	Higher train speeds result in increased vibration levels. Peak ground vibration acceleration in the time domain tends to increase approximately linearly with speed. However, the influence of speed decreases with distance from the track.	Xia et al. [21], Zhai et al. [22], Alexandrou et al. [23], Volberg [24], Yokoyama et al. [25], Mirza et al. [26], Zhai and Cai [27]
	Suspension and vehicle masses	The primary suspension system has the greatest influence on vibration levels. An increase in the unsprung masses leads to a significant rise in vibration levels. At distances close to the track, quasi-static excitations are prevalent and therefore the unsprung mass has little effect. In addition, as distance from the track increases, the frequency content exhibits much larger changes for different unsprung masses. As offset increases, these frequency changes occur over an increasingly broad band of frequencies. This is because the unsprung mass is highly influential in the generation of dynamic excitation, which becomes increasingly dominant with track offset.	Costa et al. [28], Mirza et al. [26], Kouroussis et al. [29], Colaço et al. [30], Zhai and Cai [27]
	Axle and bogie spacing	Vibratory response is amplified at specific frequencies associated with the spacing between axles and bogies.	Kouroussis et al. [29], Mirza et al. [26], Milne et al. [31]
	Wheel conditions	Irregular or out-of-round wheels significantly influence vibration levels by increasing dynamic wheel–rail interaction forces. These effects are particularly pronounced in the medium- to high-frequency range.	Kouroussis et al. [32], Alexandrou et al. [23], Nielsen et al. [33], Mosleh et al. [34]
Track	Rail irregularities	Rail surface irregularities generate dynamic interaction forces between the wheel and rail. Maintaining good rail conditions helps reduce induced vibration levels.	Xu et al. [35], Sadeghi et al. [36], Grassie [37], Xing et al. [38], Clark et al. [39], Colaço et al. [40], Kouroussis et al. [41]
	Track type	Track type is a key factor influencing vibration levels. Stiffer track systems typically emit higher levels of vibration. The implementation of resilient elements, such as fasteners, mats, and floating slabs, can significantly reduce vibration. These elements function by introducing a new resonance frequency into the system, thereby improving performance at higher frequencies.	Ntotsios et al. [42], Thompson and Jones [43], Costa et al. [28]
	Track location	Railway lines can be constructed in three primary configurations: underground, at ground level, or elevated. Each configuration leads to distinct patterns of vibration generation and transmission:	Lopes [44], Eitzenberger [45], Chen et al. [46], Olivier et al. [47]
	Tunnel track: Vibrations are generated and transmitted directly into the surrounding soil mass.
	At-grade track: Both airborne noise and ground-borne vibrations are emitted.
	Elevated track: Airborne noise is produced, and vibrations can be transmitted through the supporting structure, such as a viaduct, before reaching the ground.

provides a systematic review of the literature, aimed at identifying the main factors related to rolling stock and track components that contribute to vibration generation.

From a modelling perspective, it should be noted that train dynamic loads are still commonly represented using simplified approaches, such as the moving point load [48], lumped parameter model [49], or the vertical train–track coupled dynamic model [50]. However, in recent years, increasing attention has been devoted to the development of time-domain train–track–tunnel–soil interaction models, which aim to comprehensively capture the coupled dynamic behaviour of the train–track and tunnel–soil systems. Such models significantly improve the prediction accuracy of train-induced vibrations within the tunnel–soil system, whether formulated analytically [51], semi-analytically [52,53,54], or numerically [55,56,57]. Such advancements in train–track coupled dynamics have facilitated the adoption of more realistic representations of train loading. Nevertheless, considerable potential remains for improving both computational efficiency and the accuracy of train load estimation, particularly in cases involving curved railway alignments.

From the reviewed documents, and particularly regarding the modelling of rolling stock, it is important to emphasise some conclusions reported in Colaço et al. [30]. In many cases, a complete characterisation of the mechanical parameters of a vehicle is not feasible. This limitation arises from several factors, including the lack of data provided by railway operators, the restricted availability of information from manufacturers, and the inherent difficulties associated with experimental testing. The study concluded that the adoption of simplified numerical models, in which only the unsprung masses are considered, is suitable for the prediction of ground-borne vibrations.

From another perspective, track irregularities play a decisive role in vibration levels. Different approaches for defining the power spectral density (PSD) of track irregularities can be found in the literature, as reported by Kouroussis et al. [41]. After analysing some of these approaches, Colaço et al. [40] concluded that the selection of an appropriate PSD is a critical step. Indeed, prospective studies of ground vibrations carried out without specific information on rail irregularities may lead to markedly different results, depending on the PSD function adopted and the assumed track condition. It is therefore essential to remain aware of this limitation in order to support reliable engineering judgments.

3.2. Propagation Through the Soil and Structure

Vibrations generated at the source are transmitted into the surrounding ground, giving rise to the propagation of elastic waves through the soil. In the case of vibrations induced by railway traffic, the strain levels involved are generally low enough for the soil to be modelled as a linear viscoelastic medium (see

Table 8. Validity range of constitutive models as a function of the shear strain (adapted from [58]).

Shear Strain	10−6	10−5	10−4	10−3	10−2	10−1
Phenomenon	Wave propagation, vibrations			Cracking, differential settlements	Slippage, compaction, liquefaction
Mechanical characteristics	Elastic			Elastoplastic	Failure
Properties	Shear modulus, Poisson’s ratio, material damping				Friction angle, cohesion
Simulation models	Linear (visco) elastic model			Equivalent linear viscoelastic model	Non-linear cyclic model

). This assumption simplifies the analysis and enables the use of linear wave propagation theories to model vibration transmission.

The vibrations travel through the propagation medium until they reach a receiver, such as a building. The propagation path is influenced by a variety of factors, with geotechnical properties playing a particularly significant role. The most influential geotechnical parameters include the mechanical properties of the soil, especially stiffness and damping ratio, as well as stratification, depth to bedrock, or the position of the groundwater table. A systematic literature review summarising the key factors influencing ground-borne vibration propagation is presented in

Table 9. Factors in the propagation path influencing vibration levels.

Component	Factors	Influence	References
Ground	Mechanical properties	Vibration levels generally increase as soil stiffness decreases. Nevertheless, at larger distances, vibration levels in both softer and stiffer soils tend to converge and become comparable.	Kouroussis et al. [59], Gupta et al. [60], Lopes et al. [61]
	Depth to bedrock	Vibration levels tend to attenuate more rapidly with distance from the track when propagating through soil than through rock. This effect is more pronounced in the presence of a strong stiffness contrast between soil layers.	Kouroussis et al. [59]
	Stratification	Soil stratification can have a substantial impact on vibration levels, as the dynamic properties of individual layers may differ significantly. These variations can result in the amplification of vibrations at certain frequencies, depending on the layering profile.	Kouroussis et al. [59], Thompson et al. [62]
	Water table level	The water table can have a substantial impact on vibration transmission. Vibration levels in saturated soils are lower than in unsaturated soils due to the incompressibility of the saturated medium.	Schevenels et al. [63], Li et al. [64], Bayindir [65], Gupta et al. [60]

.

The mechanical properties of the track play a crucial role in determining vibration levels. An extensive investigation of this subject was carried out by Kouroussis et al. [59]. As a first approach, and assuming a homogeneous ground, the study shows that vibration levels decrease as the shear modulus and damping increase. When a more realistic case of a layered ground is considered, the authors examined several specific configurations:

• A soil layer overlying a stiffer substratum;
• A layered ground where rigidity increases with depth;
• A layered ground where rigidity decreases with depth.

From these representative scenarios, a number of general conclusions were drawn, which are summarised in

Table 10. General conclusions about the trends of ground vibration for different geotechnical scenarios [59].

Scenario	Vibration Level
	Homogeneous ground
Shear modulus sensitivity	The vibration level diminishes with increasing stiffness.
	Layered ground
Soil layer over bedrock	An important decrease when the top layer height is relatively small.
Rigidity increasing with depth	A relative decrease with the number of layers.
Rigidity decreasing with depth	A relative increase with the number of layers.

.

3.3. Dynamic Response of the Receiver

The wide variety of building typologies, coupled with the dynamic interaction between soil and structure, makes the analysis of ground-borne noise and vibrations within buildings a highly challenging task. This analysis involves modelling two distinct media, soil and structure, with distinct mechanical properties and behaviours. When subjected to an incident wave field, the interaction between these two media gives rise to two main effects: kinematic interaction and inertial interaction, as schematically represented in Figure 5.

Kinematic interaction refers to the disturbance of the free-field ground motion caused by the presence of structural elements with stiffness different from that of the surrounding soil. This leads to wave scattering, as rigid foundations are generally unable to follow the ground deformations induced by seismic waves.

In contrast, inertial interaction arises when the structure, subjected to vibrations at its base, generates inertial forces that in turn induce additional deformations in the surrounding soil. This mutual movement between soil and structure characterises the inertial component of the interaction [66].

Figure 5. Free-field motion and its relationship to kinematic interaction and inertial interaction [67]. ($u_g$—free-field motion; $u_{FIM}$—foundation input motion; ${\theta }_{FIM}$—foundation rotation).

Figure 5. Free-field motion and its relationship to kinematic interaction and inertial interaction [67]. ($u_g$—free-field motion; $u_{FIM}$—foundation input motion; ${\theta }_{FIM}$—foundation rotation).

Generic adjustment factors are often applied to free-field vibration levels to account for soil-structure coupling effects, the attenuation/amplification of vibrations within the building and the conversion of vibration levels into noise levels [11]. However, for more accurate assessments, numerical simulations are typically used to predict vibration responses at specific locations within the structure.

Table 11. Receiver characteristics influencing vibration levels.

Component	Factors	Influence	References
Building	Foundation	The greater the mass of the building applied to the foundations, the stronger the dynamic coupling between the soil and the foundations, which leads to a reduction in vibration levels within the building.	Hussein et al. [68], Coulier et al. [66], FTA [11]
	Construction characteristics	The largest displacements in structural elements typically occur at their resonant frequencies. Vibrations associated with the lowest natural modes of slabs often lead to maximum displacements at mid-span. The natural frequencies of slabs depend on their geometry, stiffness, and support conditions.	Auersch [69], Kuo et al. [70], López-Mendoza et al. [71]
	Number of building floors	Vibration levels generally decrease with height, decreasing progressively from the base to the upper floors. However, in buildings with low mass, this attenuation can be minimal or even negligible. In certain cases, vibration levels may actually increase on higher floors due to structural amplification effects.	Quagliata et al. [11], Xia et al. [21], Hanson et al. [12], Kurzweil [72], Auersch [69]
	Acoustic response	Ground-borne noise is typically classified as low-frequency noise, generally occurring below 250 Hz.	Colaço et al. [73,74], Ghangale et al. [75], Nagy et al. [76], Fiala et al. [77,78]

provides general considerations regarding the key factors that influence the dynamic behaviour of buildings.

As mentioned in Table 11, the receiver response is strongly influenced by building properties. A key factor in this context is the amplification effect of floors. Studies by Kuo et al. [70] and López-Mendoza et al. [71] provide clear evidence of the significant role of floor amplification in shaping the vibration levels perceived by inhabitants. The dynamic coupling interaction between the soil and the structure must also be considered. Both authors suggested the adoption of a global factor that accounts for slab amplification effects and soil–structure coupling. Since both effects are frequency-dependent, the computation of a transfer function for buildings over the relevant frequency range is essential, highlighting the importance of building properties in the assessment of vibration perception.

4. Prediction Methods

Although experimental vibration tests provide a practical and highly accurate means of assessing vibration levels, there are circumstances in which such methods are impractical. For instance, limited site accessibility, high background noise in urban settings, and complex configurations of tunnels and buildings can pose significant challenges to their implementation [79].

As an alternative, railway-induced noise and vibration can be predicted using a range of prediction approaches. These approaches are generally classified into five main categories: analytical and semi-analytical, empirical models, numerical models, artificial intelligence (AI)-based approaches, and hybrid models. In the following sections, the current state of the art in each of these categories is comprehensively reviewed.

4.1. Analytical Models

To estimate ground vibration levels generated by underground trains, Metrikine et al. [80] developed a two-dimensional (2D) analytical model, treating the tunnel as an infinitely long Euler–Bernoulli beam embedded in a viscoelastic soil layer over a rigid base. While this assumption is reasonable for long tunnels, deviations may occur in cases involving short or curved tunnels due to boundary and geometric effects. Comparative study with 3D models [81] suggests that although 2D models provide results that qualitatively agree with those of 3D models at most frequencies, such simplifications typically lead to significant discrepancies in certain frequencies. Later, this model is refined by introducing a half-space ground beneath the beam and employing a wavelet-based approach to perform the inverse Fourier transform, thereby converting the responses from the wavenumber domain to the spatial domain [82]. Given that 2D analytical embedded beam models cannot fully capture the complex dynamics of real tunnel structures, there is a clear need for the development of a three-dimensional (3D) underground railway model. Such a model should incorporate the dynamic interactions between the train, the track, the tunnel, and the surrounding soil, while retaining the computational and conceptual advantages of an analytical framework.

Several years later, He et al. [83] developed a general analytical solution for a tunnel embedded within a multi-layered half-space. In their formulation, the tunnel is idealised as an infinitely long, elastic, hollow cylinder, while the surrounding soil is represented as a stratified half-space composed of multiple layers.

4.2. Semi-Analytical Models

One of the most prominent semi-analytical models employing a two-and-a-half-dimensional (2.5D) approach is the Pipe-in-Pipe (PiP) model [84,85]. This model offers a computationally efficient means of analysing underground railway tunnels with circular cross-sections embedded within an unbounded medium. In the PiP model, the surrounding soil is represented as a 3D, homogeneous, isotropic elastic solid, idealised as a thick-walled cylinder. The cylinder has an inner diameter matching that of the tunnel and an outer diameter extending to infinity, as illustrated in Figure 6a.

An extension of the PiP model to a layered half-space was later introduced by Hussein et al. [86], employing a fictitious force method. This approach provides accurate results when the tunnel is located sufficiently far from the free surface or adjacent layer interfaces. A schematic illustration of this model is presented in Figure 6b. Additionally, it demonstrates a substantial improvement in computational efficiency compared with the coupled FE–BE model; in particular, this method has been observed to be nearly 20 times faster than the coupled FE–BE approach.

Most recently, He et al. proposed a novel 2D semi-analytical method to compute ground vibrations generated by a circular tunnel in both a homogeneous half-space [87] and a layered half-space [88] with an irregular surface. The method enables the simulation of topographies with arbitrary shapes by discretising only the complex, irregular portion of the ground surface. This characteristic enhances the computational efficiency of the approach, making it a robust tool for investigating the propagation of train-induced vibrations across complex ground. Nevertheless, these semi-analytical approaches are limited to very simple tunnel geometries, making them unsuitable for modelling complex tunnel-soil interactions or irregular configurations. Therefore, numerical alternatives, whether mesh-based or meshless, are typically required to handle cases involving complex geometries. Comparisons of semi-analytical models against measurement data demonstrate that these approaches achieve very accurate performance in predicting railway-induced vibrations [89].

4.3. Numerical Models

4.3.1. Mesh-Based Approaches

Various mesh-based numerical modelling strategies have been proposed to simulate train-induced vibrations originating from underground tunnels, either in 2D or 3D [90,91,92,93,94]. A schematic representation of the 2D and 3D finite element method (FEM) meshes used for modelling underground railway tunnels is shown in Figure 7 [92]. In this framework, a modular approach was developed to estimate these vibrations, in which a 3D FE model of the tunnel is first used to derive equivalent Timoshenko beam parameters. To properly simulate the unbounded domain and mitigate spurious wave reflections in the FEM model, viscous boundaries are employed. These parameters are then employed in a 2D side-view beam model of the track, coupled with a mass–spring representation of the train, to calculate the under-sleeper forces [95]. To improve the computational efficiency of full 3D FE simulations, a vehicle–track–tunnel–soil model was later introduced to predict vibrations and stresses induced by wheel–rail roughness in the tunnel–soil system [54]. Another approach utilises a 3D coupled finite element–boundary element model, where the tunnel is modelled using the FEM and the surrounding soil with the boundary element method (BEM), enabling efficient and accurate prediction of ground-borne vibrations from underground railways [81,96]. Both at-grade and underground railway tunnel systems are often assumed to be longitudinally invariant. This characteristic enables the use of 2.5D modelling approaches to assess soil–structure interaction (SSI). A schematic illustration of the 2.5D SSI approach is shown in Figure 8. In recent years, 2.5D modelling has gained increasing attention due to its advantages over fully 3D models, including reduced computational cost and complexity while retaining sufficient accuracy for many practical engineering applications.

The applicability of 2.5D modelling strategies to SSI problems was first investigated by Hwang et al. [97], who presented the initial work on 2.5D FEM, along with an application of the proposed methodology to soil-structure systems. Later, Yang and Hung [98] developed a 2.5D approach based on finite and infinite elements to model longitudinally invariant unbounded systems subjected to moving loads. In their approach, the infinite elements are used to represent the unbounded domain. Furthermore, a 2.5D FEM approach was introduced by Gavríc to compute the dispersion curves associated with longitudinally invariant structures with thin-walled [99] and solid [100] cross-sections.

A well-established method for addressing SSI problems is the coupled FEM-BEM approach, where the FEM models the structure, and the BEM is used to account for the soil medium. Sheng et al. [101,101] proposed a FEM-BEM approach within the 2.5D domain. François et al. [102] also introduced a 2.5D FEM-BEM approach that employs the Green’s functions of a layered half-space as fundamental solutions, rather than full-space ones, significantly reducing the number of boundaries to be meshed. The application of this method to predict railway-induced vibrations is explored in [103].

An alternative method for obtaining the Green’s functions required in 2.5D BEM in elastodynamics is the thin layer method [104]. Another approach to address SSI problems was proposed by Lopes et al. [105], who studied the vibrations induced in buildings by underground railway traffic using the 2.5D FEM coupled with the Perfectly Matched Layer (PML) technique. In this method, the FEM is used to model the structure, while the PML accounts for the unbounded soil domain. PML is a layer of finite thickness placed beyond the boundary, designed to absorb outgoing waves and prevent reflections at the model boundaries.

In 2.5D models, the structure and surrounding soil are assumed to be invariant in the longitudinal direction. However, due to the physical periodicity of the railway tracks, this assumption does not hold for all frequencies. Specifically, for typical railway tunnel structures, the periodicity of the system does not influence the transfer functions at frequencies below 80 Hz [106]. Alternatively, periodic modelling can be employed to simulate the response of railway systems at higher frequencies. In this regard, Gupta et al. [107] compared the results obtained from a coupled FEM-BEM periodic model of the track-tunnel-soil system with those from the PiP model, highlighting the benefits and drawbacks of both models. Furthermore, Gupta and Degrande [106] evaluated the efficiency of continuous and discontinuous floating slab tracks using a periodic model of the track-tunnel-soil system. When mesh-based numerical methods are compared with experimental measurements, discrepancies of around 3 dB have been reported when using 2D FEM [108], 3D FEM [93], and approximately 3 dB when employing 2.5D FE–BE [109] for the response obtained at the free field.

To overcome the limitations of mesh-based approaches and enhance the computational efficiency of integration-based numerical strategies, meshless methods have been proposed to model wave propagation.

4.3.2. Meshless Approaches

Meshless methods, as an alternative to mesh-based approaches, have gained increasing attention from the research community over the past few decades. In meshless methods, there is no inherent dependence on a specific mesh topology, which leads to simpler formulations and computational implementation procedures. In recent years, numerous studies have explored novel meshless approaches. The Method of Fundamental Solutions (MFS), Singular Boundary Method (SBM) and Generalised Finite Difference Method (GFDM) are perhaps the most widely used methods in this category. In the following, a general conceptual description of these two methods is provided, along with a review of previous studies in the literature. A summary of alternative meshless methods is also presented thereafter.

The MFS is a meshless method that uses the fundamental solution of the governing equation of interest as the interpolation basis function (Figure 9b). It is particularly effective for solving wave propagation problems in unbounded or partially unbounded domains. This method offers two main advantages over the BEM in terms of numerical efficiency: firstly, it does not require discretising the boundary or performing integration over it, and secondly, the system of equations to solve is typically much smaller than the one required in BEM. The MFS relies on a distribution of collocation points, which evaluate the boundary conditions at discrete locations, and a set of source points (or virtual forces in elastodynamic problems), where the strengths of sources are calculated to satisfy the boundary conditions at the collocation points. In the MFS, the collocation points are placed on the boundary, while the source points are positioned outside the domain.

To extend the capabilities of the MFS, methodologies combining it with mesh-based approaches to address SSI problems have been developed. Godinho et al. [110] presented a 2D FEM-MFS modelling approach for such problems. An extension of this method to the 2.5D domain was proposed by Amado-Mendes et al. [55], where a methodology is introduced that models the structure using 2.5D FEM and the surrounding soil with 2.5D MFS. Godinho et al. [111] also presented a fully meshless method to solve SSI problems in the frequency domain, where the MFS is used to model the soil and the meshless local Petrov–Galerkin method is employed to model the structure. More recently, Liravi et al. [112] proposed a 2.5D FEM-BEM-MFS approach, in which the BEM is used to obtain the soil stiffness matrix for SSI, while the MFS is employed to determine the radiated field in the soil induced by the system. This work also introduces a control methodology to reduce the errors associated with MFS predictions. However, both the FEM-MFS and FEM-BEM-MFS methods face challenges when dealing with complex boundary shapes, due to the difficulties in selecting an appropriate distribution of virtual sources for these geometries.

The SBM is a novel and emerging meshless boundary collocation method for solving boundary value problems that, unlike the MFS, locates the virtual source points on the physical boundary. The SBM inherits key advantages from both the BEM and the MFS. On the one hand, it avoids the computationally expensive integration procedure required in BEM-based methods. On the other hand, the SBM addresses the limitations of the MFS related to the positioning of the virtual source points by placing them on the physical boundary, overlapping the collocation points. While this technique eliminates the challenges associated with the distribution of source points, it introduces singularities in the fundamental solutions on the boundary due to the overlap between collocation points and virtual forces. To overcome these singularities, a regularisation technique is necessary. In the SBM, this is achieved by introducing the concept of the origin intensity factor (OIF) [113].

The SBM has been extensively explored for the prediction of railway-induced noise and vibration in previous studies. The FEM–SBM methodology represents the first attempt to employ SBM in this context, where the FEM is used to model the structural components, and SBM is applied to model wave propagation in the surrounding medium in both 2.5D [114] and 3D [115] formulations. To enhance the accuracy of SBM for highly complex geometries, it was subsequently combined with the MFS, resulting in a novel approach called 2.5D SBM–MFS, as shown in Figure 9c. In this method, the SBM is employed to handle the complex portions of the geometry, while the MFS is applied to the smoother regions. This hybrid method has proven to be a promising alternative to conventional SBM techniques for simulating both elastic [116,117] and acoustic wave propagation [118,119]. A schematic illustration of the MFS, SBM, and combined SBM–MFS approaches is provided in Figure 9.

The GFDM corresponds to an advancement of the classical finite difference method, founded on Taylor series expansions and the moving least squares technique. This meshless approach eliminates the need for numerical integration and is free from the constraints of a predefined mesh. The core idea of the GFDM lies in approximating the derivatives of unknown variables as a linear combination of their values at neighbouring nodes within a local stencil, often referred to as a star. Furthermore, the coefficient matrix arising from the GFDM is typically sparse, which contributes to substantial computational efficiency when compared to the dense matrices commonly generated by boundary-type meshless methods [120]. In evaluating the performance of meshless approaches against experimental measurements, it has been observed that the 2.5D FEM–MFS method is able to predict the vibration velocity amplitude levels at the free field with good accuracy [121].

4.4. Empirical Models

Despite substantial recent progress in the development of numerical models for railway-induced ground vibrations, their practical application remains predominantly within the realm of academic research. In contrast, empirical methods continue to prevail in engineering practice because of their simplicity and low associated engineering costs [122]. When significant uncertainties exist in model parameters, numerical approaches may struggle to yield reliable results. In such cases, empirical methods can offer a practical alternative for estimating the propagation of vibrations and noise from underground railways. Examples of empirical methods include those developed by the FRA and the FTA of the U.S. Department of Transportation [11,12], the Swiss Federal Railways (SBB) [123], the method proposed by Madshus et al. [124] based on measurements conducted in Norway and Sweden, and the approach by Hood et al. [125], which was developed within the framework of the Channel Tunnel Rail Link project in the UK. The procedures established by the FRA and FTA align with the three assessment levels defined in the ISO 14837-1 standard [126]. The Detailed Vibration Assessment, in particular, is based on prediction techniques developed by Bovey [127] and Nelson and Saurenman [128], offering a more comprehensive methodology for predicting ground-borne vibrations and re-radiated noise within buildings. The SBB method differentiates between two models: VIBRA-1 and VIBRA-2. The latter provides a more detailed analysis by incorporating, for example, frequency-dependent attenuation models. Similarly, the empirical approaches in [124,125] follow a comparable structure to that of the SBB method, while also addressing the issue of prediction uncertainty. Nevertheless, it should be noted that empirical models are generally not highly accurate. For example, considerable discrepancies have been reported between measurements and vibration predictions obtained using the FTA model, owing to its limitation of employing a single base curve for the assessment of vibrations in both at-grade railways and underground systems [36].

4.5. AI-Based Models

Although numerical analysis methods can predict vibrations with high accuracy, their drawbacks include complex models, long computation times, and limited practicality. In recent years, there has been growing interest in the application of data-driven and machine learning (ML) approaches for modelling and predicting train-induced ground-borne vibrations. These methods offer promising alternatives to conventional numerical models, particularly in scenarios where field data are available but physical models may be computationally expensive or uncertain due to subsurface variability.

Traditional ML algorithms, such as support vector machine (SVM), artificial neural network (ANN), and random forest (RF), among others [129,130], have been extensively explored for the prediction of ground-borne vibrations. In early studies within this framework, SVMs combined with wavelet transforms were employed to predict train-induced vibrations in multi-storey buildings [131]. Subsequently, ANN models were developed to estimate ground vibration amplitudes caused by urban railways [132] and high-speed trains [133]. More recently, an RF model optimised using Bayesian techniques was proposed to analyse vibration data from elevated high-speed railways [134].

Beyond traditional ML, deep learning techniques have been employed to analyse and predict ground vibrations induced by railway systems. A deep learning-based model has been utilised to identify ground-borne vibrations generated by metro trains, incorporating an uncertainty quantification process to enhance the reliability of the predictions [135]. Furthermore, it is considered a promising approach to train deep neural network (DNN) models using both simulated and measured vibration data via transfer learning (TL) strategies. This enables the model to leverage the strengths of both datasets, thereby improving the accuracy and robustness of vibration predictions [136]. However, it should be noted that, according to previous studies [137], the train–test split ratio has a significant influence on the classification performance of the three pre-trained networks. In most cases, ratios of 80–20% and 90–10% yield better results. Moreover, track-based measurements can be used to infer ground response, where deep learning techniques based on rail acceleration signals are applied to estimate track vibration characteristics [138].

In recent years, physics-informed neural networks (PINNs) have transformed the way that partial differential equations are solved by embedding the underlying physical laws directly into the training process of neural networks. This approach enables the model to learn solutions that are consistent with the physics of the problem, even in the absence of extensive training data. PINNs have demonstrated strong robustness to noise in the data, achieving reasonable identification accuracy even with noise corruption levels of up to 10% [139]. Moreover, PINNs offer the advantage of strong generalisation in data-scarce regimes [140]. In the context of elastodynamics, PINNs have proven particularly effective for modelling elastic wave propagation, offering mesh-free formulations that bypass the complexities of conventional numerical discretisation. By working in the frequency domain, this method captures wave behaviour with high accuracy while significantly reducing computational overhead compared to traditional PINNs [141]. Moreover, advancements in PINN frameworks have introduced capabilities to simulate wave fields in complex media without relying on labelled data, addressing one of the major challenges in data-scarce environments [142].

The uncertainty quantification of ML-based models has also gained attention. Liang et al. [143] integrated Bayesian neural networks with deep learning to assess uncertainties in vibration predictions. Wu et al. [144] used probabilistic methods to analyse metro-induced vibrations, offering insights into reliability analysis under varying operational conditions. It is worth noting that the uncertainty of prediction models can be further reduced by incorporating field data for the training and validation of ML models [145,146].

Surrogate models have been employed to support vibration prediction, particularly in the contexts of optimisation and uncertainty analysis. Two recent examples involve the integration of a Bayesian optimisation framework with surrogate modelling for underground railway tunnel design [147] and acoustic barriers [148], illustrating the synergy between data-driven prediction and design optimisation. In this approach, RF regression is used as the surrogate model, and the overall framework is illustrated in Figure 10 [147]. Assessing the performance of the RF model in predicting railway-induced vibrations, in comparison with measurement data, demonstrates its capability to provide accurate and instantaneous predictions [134]. Another application is the development of efficient scoping models to aid planners during the early design stages, providing rapid vibration assessments across varying soil conditions near high-speed rail lines [149,150].

The literature reflects a significant transition from conventional physics-based simulations to hybrid and data-driven approaches. While physics-informed and deep learning models offer flexibility and predictive potentiality, challenges remain in model interpretability, generalisation to unseen scenarios, and integration with field data.

4.6. Hybrid Models

The uncertainty in railway-induced ground-borne vibration predictions in buildings, primarily due to limited knowledge of local subsoil conditions, has been highlighted in numerous studies. This challenge can be addressed by integrating field data with numerical models, thereby leveraging the strengths of both approaches [122,151,152,153,154]. A variety of hybrid modelling techniques have been developed to predict ground-borne vibrations in buildings resulting from railway activity, particularly in the presence of SSI. One such approach integrates pre-calculated results from detailed numerical simulations, an experimental database compiled from multiple field measurements, and specific analytical models [155]. Another method represents buildings using finite axial rod elements, with floor impedance derived from infinite thin plate theory, while the incident wave field is characterised by either known column base forces or measured vibrations at key interface locations [156]. In addition, a virtual moving source can be utilised to simulate building vibrations through back-analysis procedures, significantly reducing the influence of uncertain parameters on prediction accuracy. This technique combines the versatility of numerical simulations with the realism of field measurements [157]. Another notable hybrid framework integrates recorded data with numerical predictions, following the source–path–receiver decoupling strategy as defined by the FTA. This method effectively separates the vibration generation, propagation, and reception mechanisms for improved analysis and interpretation [70]. When such hybrid models are compared with measurement data, they are generally found to be accurate, with discrepancies between the numerical model and experimental results typically reported to be below 4 dB [158,159].

To improve computational efficiency, a modal superposition-based model has been introduced to estimate railway-induced ground-borne vibration levels in buildings, particularly under scenarios where the incident wave field is known, either through numerical simulations or ground surface measurements. This approach incorporates SSI effects by introducing spring-damper elements at the building foundation [160]. An extension of this method includes a more rapid model employing a 3D time-domain FEM for structural modelling, alongside enhanced spring-damper representations to better capture SSI effects [71]. Additionally, the use of meshless methods as the numerical component of hybrid models offers a promising alternative that not only reduces uncertainties in the predictions but also improves computational efficiency [121,122,158,159], as illustrated in Figure 11. For the meshless approach, the continuity of displacement and the equilibrium of forces at the interface are employed to couple the FEM and MFS approaches. The soil–foundation numerical model is experimentally validated by comparing the numerical transfer functions with the corresponding experimental measurements [159].

While previous hybrid models typically combine semi-analytical or numerical simulations with field measurements, the computational cost of these approaches often remains high. However, real-time prediction capabilities are essential in certain scenarios, particularly when environmental vibrations may impact sensitive structures. To address this, recent developments have focused on hybrid strategies that enable real-time prediction while simultaneously reducing the uncertainty associated with vibration modelling by using a deep learning technique [161] or multi-point transfer ratio of vibration response [162].

4.7. Overview of Prediction Methods

In this section, the advantages and disadvantages of the previously reviewed prediction methods are summarised from a general perspective in

Table 12. Comparison of modelling approaches for railway-induced vibrations.

Method	Advantages	Disadvantages	References
Empirical models	Simple and fast to apply; low computational cost; useful for scoping studies and regulatory assessments.	Limited accuracy for complex geometries and heterogeneous soils; strongly dependent on available measurement data; poor extrapolation beyond calibrated cases.	[43,163]
Semi-analytical/analytical models	Provide physical insight into wave propagation and soil–structure interaction; lower computational cost than full numerical models; suitable for parametric studies.	Require simplifying assumptions (e.g., homogeneous soil, linearity, infinite/regular geometry); accuracy decreases in complex or strongly heterogeneous conditions.	[41,164]
Numerical models	High fidelity; capable of handling complex geometries, layered soils, and non-linear behaviour; can model both frequency- and time-domain responses.	Computationally expensive; require detailed material and soil data; sensitive to boundary conditions and meshing strategies.	[165,166,167]
AI/Data-driven models	Once trained, allow very fast predictions; can approximate highly non-linear behaviour; enable rapid scenario exploration and optimisation; useful as surrogates for numerical simulations.	Require large and representative training datasets; generalisation outside training domain may be poor; reduced interpretability compared to physics-based models.	[77,168,169]

.

5. Mitigation Measures

5.1. Overview

Based on a solid understanding of the phenomena and the modelling approaches previously discussed, it becomes possible to identify effective solutions for their mitigation, particularly with regard to structural vibrations and the associated noise. In problems of this nature, mitigation strategies can be categorised according to the location at which they are implemented. Accordingly, GBN and GBV can be controlled at three distinct levels: at the source, along the transmission path or at the receiver.

Despite considerable progress in the study of mitigation measures, a holistic approach of the entire system, accounting for both GBV and GBN, remains essential. Only through such a holistic perspective, the effectiveness of mitigation strategies can be properly assessed. It is important to note that, in some cases, certain solutions may reduce the problem within specific frequency ranges while significantly amplifying them in others. Moreover, combining multiple mitigation measures across different levels presents an additional challenge that can be addressed through an integrated approach.

5.2. Mitigation Measures at Source

For the purpose of the present section, the vibration source domain comprises both the vehicle and the track. The interaction between these components constitutes a highly complex dynamic system. A comprehensive analysis of this system is essential to gain a deeper understanding of the mechanisms responsible for vibration generation at the source. Such insight is crucial for the design and development of more effective and targeted mitigation strategies.

Table 13. Mitigation measures of ground-bone vibrations applied at the source.

Location	Component	Measures	References
Vehicle	Wheelsets	Good maintenance practices.	Nielsen et al. [170,171], Satis et al. [172], Ju et al. [173], Suarez et al. [174], Kouroussis et al. [168,175], Wilson et al. [176], Colaço et al. [30], RIVAS [33]
	Unsprung masses	Improvement of suspension systems and reduction in unsprung masses.
	Operating speed	Decrease in vehicle speed.
	Resilient wheels	Replacement of conventional wheels with resilient wheels.
	Dynamic Vibration Absorbers (DVA)	Design and installation of DVAs on the vehicle bogie.
Track	Rail	Proper maintenance or replacement of the rail. Use of embedded rail systems. Installation of rail dampers.	Marshall et al. [177], Lakušić et al. [178], Bentn et al. [179], Hanson et al. [12], Satis et al. [172], Bongini et al. [180], Nelson et al. [181], Grootenhuis et al. [182], Hemsworth et al. [183], Wang et al. [184], Costa et al. [185], Dahlberg et al. [186], Ferro et al. [187], Mateus et al. [188], Shamayleh et al. [189], Auersch [190], Wang et al. [191], He et al. [192]
	Rail fastening system	Use of fasteners with resilient components to prevent direct contact between rail and sleeper.
	Sleepers and ballast	Use of composite or wooden sleepers, installation of resilient elements beneath sleepers, installation of ballast mats.
	Track type	Increase in ballast layer thickness, installation of floating slab tracks, increase in tunnel depth.
	Dynamic Vibration Absorbers (DVA)	Design and placement of DVAs at specific rail defect locations.

summarises a set of vibration mitigation measures applied at the source level.

When addressing mitigation measures at the source, a common solution, particularly in urban environments, is the use of a resilient mat, either beneath the slab in slab track systems or beneath the ballast in ballasted tracks. As indicated by Costa et al. [185], the efficiency of these elements depends strongly on their stiffness and exhibits a clear frequency-dependent behaviour. Moreover, the inclusion of mats influences not only the vibrations transmitted to the ground but also the dynamic behaviour of the track itself and, consequently, the train–track interaction loads. Since most studies on the efficiency and design of such measures rely on receptance and mobility concepts, without accounting for changes in the overall compliance of the train–track system, the reported findings are of substantial practical relevance.

5.3. Mitigation Measures at Propagation Path

In situations where it is not feasible to implement mitigation measures directly at the source to improve its condition and reduce vibration generation, alternative strategies can be applied along the propagation path. The primary objective of these measures is to change the transmission characteristics of elastic waves as they travel through the ground.

Interventions along the propagation path are typically implemented as close as possible to the vibration source and are, therefore, most commonly applicable to at-grade railway infrastructures. These measures work as barriers to wave propagation, placed between the source and the receiver, with the aim of reducing or redirecting the transmitted elastic waves.

Table 14. Vibration mitigation measures applied at the propagation path.

Component	Measures	References
Embankment and platform	Increasing the height of the embankment or reinforcing the foundation soil of the platform aims to create a wave-guiding effect, thereby conditioning wave propagation through the medium.	Lakušić et al. [178], Olivier et al. [47], Connolly et al. [193], Lyratzakis et al. [194,195]
Barrier	The presence of obstacles along the propagation path provides scattering and/or damping effects on incident waves. This effect can be achieved by introducing materials or systems specifically designed for this purpose into the ground, leveraging their mass, stiffness or geometric configuration. Examples of such mitigation measures include trenches and barriers, large mass elements, soil stiffness enhancement techniques, and, more recently, metamaterials.	Çelebi et al. [196], Thompson et al. [197], Yang et al. [198], Jayawardana et al. [199], Guo et al. [200], Ahmad and Al-Hussaini [201], E Richart [202], Beskos et al. [203], Barbosa et al. [204], Hassoun et al. [205], François et al. [206], Coulier et al. [207,208,209], Kattis et al. [210], Takemiya et al. [211], Krylov et al. [212], Dijckmans et al. [213,214], Adam et al. [215], Ma et al. [216], Barbosa et al. [204], Jones [217], Kouroussis et al. [218], Castanheira-Pinto et al. [219], Albino et al. [220]
Soil	Enhancing the mechanical properties of the soil has a direct impact on vibration propagation. Increasing soil stiffness can significantly reduce vibrations in the corresponding shadow zone. The effectiveness of this measure depends on the stiffness contrast between the natural soil and the improved zone, as well as the thickness of the treated layer.	Takemiya [211], Coulier et al. [207], Thompson et al. [62], Tang et al. [221], Barbosa et al. [204]

presents some relevant mitigation strategies that can be implemented along the propagation path.

As evidenced in Table 14, a wide range of solutions exist for mitigating vibrations along the propagation path. A common approach is the construction of a trench parallel to the railway line, aimed at protecting the structures located behind it. A current topic of investigation concerns the effectiveness of filling the trench, as well as the influence of its depth and position relative to the track [197,204]. Although open trenches generally provide the best results across different scenarios, their practical applicability is limited. Conversely, the stratification and stiffness of the soil in a given context have a significant influence, which restricts the generalisation of conclusions regarding trench depth and track distance. Therefore, an appropriate design should be prioritised for each specific case.

5.4. Mitigation Measures at Receiver

The receiver is defined in this study as the part of the system where vibrations generated by railway traffic are perceived. Traditionally, this issue was addressed primarily in buildings with high sensitivity to vibrations, such as hospitals, research facilities or concert halls. In such cases, mitigation measures were integrated into the construction or rehabilitation process to ensure compliance with service performance requirements.

However, in recent years there has been growing awareness of the adverse effects of prolonged and repeated exposure to vibration, alongside with regulatory requirements from licensing authorities for buildings located near railway lines. Simultaneously, the increasing availability of a wide range of technological solutions has led to the broader adoption of vibration mitigation measures in a more diverse set of buildings. These interventions not only ensure high service standards but also help to preserve or even to enhance property value, even when located in close proximity to vibration sources such as railways.

Table 15. Vibration mitigation measures at the receiver.

Location	Component	Measures	References
Building	Foundations	Thickening of foundation slabs; local improvement of the foundation soil; vibration isolation of the superstructure using springs or elastomeric supports.	Fiala et al. [78,222,223], Lopez et al. [224], Talbot et al. [225], Ulgen et al. [226], Yang et al. [227], Persson et al. [228], Statis et al. [172], Wilson et al. [229], Soares et al. [159], Reynolds et al. [230]
	Slabs	Reduction in the flexibility of the building slabs; active mass damping of floors.
	Room	Vibration isolation of a room or part of the structure using a box-in-box system.

summarises a range of mitigation measures applicable at the receiver level to reduce ground-borne vibrations and noise.

When mitigation measures focus on protecting the building itself, only a limited number of reliable solutions are available. Among them, base isolation can be highlighted as an effective approach [159], although it requires appropriate design and specialised expertise. From a numerical perspective, Talbot et al. [225] provide important guidelines, stressing that: (i) the building and its foundation must be considered as an integrated system; and (ii) the effectiveness of isolation is highly dependent on the choice of isolation frequency, which can significantly influence performance depending on the characteristics of the excitation.

5.5. Summary

Mitigation measures can be classified into three main categories: at the source, along the transmission path, and at the receiver. For source-based strategies, the most common solutions are applied at the track level (with the possible exception of controlling operating speed, since the definition of specific vehicle characteristics is rarely feasible). These measures are generally effective in reducing vibration levels and can be implemented along the entire track or in selected sections. However, their application in operating lines is often challenging, as they typically require significant interventions in the infrastructure. According to the literature, vibration reductions from source-based strategies can range widely from 2 to 40 dB, depending on the particular technique employed [231].

Measures applied along the transmission path (e.g., trenches, barriers, etc.) can attenuate vibrations effectively within specific frequency ranges, but their performance is highly site-dependent and often limited by construction feasibility. Such mitigation measures can result in vibration reductions in the range of 3–14 dB [231].

Mitigation measures at the receiver (e.g., building isolation) directly address vibration perception. However, they are usually costly and intrusive, which restricts their use to particularly sensitive buildings. It should be noted that improvements at the receiver location can achieve vibration reductions of between 2 and 26 dB, largely depending on the technique employed [231].

Overall, each of these approaches presents both advantages and limitations, highlighting the need for integrated and optimised mitigation strategies tailored to the characteristics of each site.

6. Perspectives and Future Research

In light of recent studies in the field of railway-induced noise and vibrations,

Table 16. Emerging research trends in railway-induced noise and vibration.

Category	Research Direction	Details
Human health and perception	Combined effects of noise and vibration	Investigate cardiovascular, mental health effects, annoyance, and sleep disturbance under joint exposure to vibration and noise.
	Epidemiological realism	Explore realistic noise-vibration exposure scenarios.
Material and structural modelling	Advanced material modelling	Incorporate non-linear behaviour and fatigue in ballast, sub-ballast, and building materials under cyclic loading.
	Realistic construction detail integration	Include connection details, material heterogeneity, and SSI in simulations.
	Rod-sprung-mass model refinement	Improve high-frequency accuracy and include spatial complexity, high-order floor modes, coupling effects, and non-structural elements.
Vibration mitigation and control	Vibration control systems	Develop optimised building designs with passive/active mitigation (e.g., tuned mass dampers, active mass dampers, base isolators, viscoelastic layers).
	Tuned Rail Dampers (TRDs)	Overcome TRD limitations by developing long-term performance data and design guidelines.
	Meta-wedge and novel barriers	Refine idealised metamaterial-based countermeasures for practical and realistic implementation.
	Composite vibration isolation walls	Test alternative materials, ratios, and layering, especially for non-uniform soil conditions.
Modelling and simulation	Non-linear SSI	Extend existing linear models to non-linear dynamics for extreme events (e.g., earthquakes, explosions).
	Numerical uncertainty quantification	Incorporate track curvature, speed, bogie distances, and observation geometry in stochastic frameworks; support with field data.
	Combination of traditional prediction models with ML	Integration of the classical prediction models with AI to leverage the benefits provided by each one.
	Digital Twin integration	Use Digital Twin technologies for visualisation and modelling of noise and vibration propagation.
Experimental methods and monitoring	Measurement point design	Develop practical guidelines for optimal sensor placement in buildings; investigate impact on monitoring outcomes.
	Mode identification techniques	Validate predictive modes through ambient vibration tests; extend beyond vertical global modes.
	Full structural characterisation	Broaden vertical modal identification to capture entire building response.
Geotechnical and site-specific effects	Heterogeneous ground conditions	Address limitations of current models based on stratified, uniform soil layers by including lateral geological variability.
	Structural-foundation-soil systems	Investigate effects of structure size, foundation type, and soil conditions to inform load modelling modifications.

presents potential topics that warrant further investigation. It should be noted that the future perspective of the field is derived from the limitations of the most recently published literature and the ways in which these shortcomings can be addressed.

7. Conclusions

Ground-borne noise and vibrations caused by railway traffic continue to pose significant environmental and societal challenges, particularly in urban areas and locations with vibration-sensitive equipment. This review has highlighted the main aspects to be considered in a railway project when addressing these phenomena. The paper has been structured as a step-by-step guide, resembling a manual for practitioners and researchers in the field. The key topics and insights can be summarised as follows:

• Regulatory frameworks: Although many countries have established regulations, significant disparities remain in assessment indicators, threshold values, and applicability, limiting the prospects for a harmonised international approach.
• Generation, propagation, and reception: The fundamental mechanisms of how vibrations and noise are generated, transmitted through the ground, and perceived at the receiver have been discussed. The general tendencies identified for each influencing factor are highly valuable as a first step in the analysis of specific cases.
• Prediction methodologies: Prediction models, although increasingly sophisticated, still face limitations in accounting for complex site-specific variables. As a result, the uncertainty of predictions remains relatively high.
• Mitigation measures: A variety of solutions exist, but their effectiveness depends strongly on local conditions and system characteristics. While some general guidelines can be identified, site-specific designs should be prioritised.

Several key areas of development are expected to drive future progress in the management of railway-induced ground-borne noise and vibrations. These include advances in the lifecycle assessment of noise and vibration emissions from both railway infrastructure and rolling stock; the development of new methodologies for long-term monitoring that enable the incorporation of real data into prediction models; and the creation of more effective and adaptable mitigation measures, along with their integration into predictive tools. Achieving these goals will require stronger collaboration between engineers, researchers, infrastructure managers, and policymakers.