Research and Advances in the Characteristics of Blast-Induced Vibration Frequencies

Abstract

Engineering blasting technology is widely applied in mining operations and the construction of buildings and structures, but ground vibrations caused by explosions are a major environmental concern. Peak particle velocity (PPV), frequency, and duration are the primary parameters for blast-induced vibrations, with PPV often used to assess the safety of vibrations. However, frequency attenuation or resonance can lead to building collapse and equipment damage, and relying solely on PPV-based safety standards is insufficient to fully evaluate the safety of blast-induced vibrations. Studying dominant frequency characteristics is crucial for protecting residents, buildings, and equipment. This paper systematically discusses the safety criteria for blast-induced vibrations in various countries, summarizes the classification of dominant frequencies, and points out that there is no clear scope of application for different dominant frequencies, with varying criteria across countries and the absence of a unified standard. Furthermore, the paper analyzes the influence of blasting parameters, explosive types, and geological conditions on dominant frequency, emphasizing the lack of research on other factors such as borehole parameters, free faces, burden, charge structure, and delay time. It also proposes the need for further exploration of factors such as charge coefficients, decoupling coefficients, borehole density coefficients, and specific explosive consumption. For dominant frequency prediction, the machine learning (ML) models proposed in this study have performed excellently in multiple experiments, especially on large-scale datasets. The experimental results show that the correlation coefficients between the predicted values of the ANN and ANFIS models and the measured data are 0.95 and 0.9988, respectively, indicating high prediction accuracy. In addition, the SVM model, when predicting the dominant frequency, generally keeps the relative error within 10%, demonstrating its efficiency and accuracy in predictions. These methods fully validate the prediction capability of the proposed models, highlighting the significant advantages of ML methods in this study and providing strong support for applications in related fields. Although ML methods can significantly improve prediction accuracy, issues such as insufficient sample size and poor generalization ability may lead to reduced prediction accuracy.

1. Introduction

With the rapid advancement of modern construction, engineering blasting technology has been widely applied in various fields such as mining, urban development, and hydraulic engineering [1] (Figure 1). This technology is favored for its efficiency and cost-effectiveness. However, alongside these advantages, blasting also generates environmental problems, including ground vibrations, fly rock, and noise pollution (Figure 2), posing threats to both the ecological environment and public safety. In recent years, researchers have gained deeper insights into the mechanisms of rock fragmentation during blasting and have made significant progress in optimizing explosive performance and blasting parameter design. Despite these advancements, the utilization efficiency of explosive energy remains relatively low. Studies indicate that only about 20% of explosive energy is effectively used for rock fragmentation, while the remaining 80% dissipates as ground vibrations, fly rock, and noise [2].

Blast-induced ground vibrations are high-intensity transient energy waves [3] and are a critical factor affecting the stability of excavation zones [4]. These vibrations pose significant threats to operational safety in the mining and construction industries, while also directly impacting the quality of life for nearby residents. Moreover, they are a key consideration for regulatory agencies when establishing safety and environmental standards. Concerns have long been raised regarding whether blast-induced vibrations can cause structural issues such as cracks, tilting, or even the collapse of buildings, bridges, and other infrastructure [5]. Additionally, recent studies have indicated that these vibrations may disrupt vegetation growth and could potentially lead to deforestation in the future [6]. Therefore, it is imperative to control the impact of blast-induced vibrations on surrounding communities and infrastructure to prevent construction accidents and property damage.

The intensity of ground vibrations can be described in terms of frequency and PPV, with PPV widely adopted across various countries as a key metric for controlling and evaluating blast-induced ground vibrations [7]. In recent years, scholars have systematically investigated the propagation and attenuation laws of blast-induced vibrations through field measurements, numerical computations, numerical simulations, and theoretical analyses [8,9]. These studies have provided a scientific basis and practical design solutions for controlling blast-induced vibrations. Accurately predicting PPV is not only a critical aspect of optimizing blasting designs but also an effective measure for minimizing the potential harm to nearby residents and structures. Traditional PPV prediction methods primarily include empirical formulas, numerical simulations, and ML techniques [10,11]. Among these, empirical formulas remain the most commonly used approach. Researchers have developed various PPV empirical formulas by employing dimensional analysis and considering factors such as blasting parameters, charge structure, and geological conditions [12,13,14]. However, the predictive accuracy of these formulas is often limited due to the constrained number of parameters considered, making them inadequate for addressing complex nonlinear factors effectively.

With the rapid advancement of ML and artificial intelligence, scholars have proposed various machine learning-based methods for predicting PPV. These include single-algorithm models such as neural networks [15,16], Gaussian process regression (GPR) [17], gene expression programming (GEP) [18], support vector machines (SVM) [19], adaptive neuro-fuzzy inference systems (ANFIS) [20], and classification and regression trees (CART) [21], as well as hybrid models integrating multiple algorithms [22,23,24,25]. Machine learning-based PPV prediction models effectively capture the nonlinear relationships among multiple influencing factors, demonstrating superior performance compared to traditional empirical formulas in terms of evaluation metrics such as MSE, R², and SE. Furthermore, ML approaches leverage multidimensional data to significantly enhance the accuracy and reliability of PPV predictions. In recent years, the study of blast-induced vibration velocity has developed into a well-established system [26]. However, systematic research on blast-induced vibration frequency remains relatively limited. In many cases, even when monitored vibration velocity exceeds the permissible PPV threshold, structural integrity may remain unaffected. Conversely, when vibration velocity is well below the allowable PPV, resonance can occur if the dominant frequency of the blast-induced vibration approaches the natural frequency of the structure, potentially causing structural damage or even collapse [27,28]. Thus, safety standards based solely on PPV are insufficient for fully evaluating the safety of blast-induced vibrations. A comprehensive safety assessment must incorporate frequency-based criteria [29,30].

In recent years, scholars have gained a certain level of understanding of blast-induced vibration frequencies, and related research findings have provided valuable guidance for engineering blasting practices. However, compared to PPV, the attenuation mechanisms, influencing factors, and prediction methods for blast-induced vibration frequencies have yet to be systematically studied and developed into a comprehensive research framework. Although frequency factors are considered in the blast-induced vibration safety criteria of various countries, their application remains incomplete and requires further optimization. Blast-induced vibration frequencies are influenced by multiple factors, including the distance from the explosion source to the monitored location (R), the physical properties of the rock, and environmental conditions [31]. However, many factors affecting the dominant frequency of blast-induced vibrations are challenging to quantify. Conducting in-depth research on the characteristics of blast-induced vibration frequencies, exploring their influencing factors, and systematically improving frequency prediction methods are essential for accurately evaluating the impact of blast-induced vibrations on surrounding environments and structures. Such studies provide a scientific foundation for optimizing blasting designs, ensuring safety, and maintaining stability during construction operations.

Currently, research on blast-induced vibrations mainly focuses on PPV, systematically examining the influence of various factors on PPV and its prediction methods. However, PPV only reflects the intensity of vibrations and cannot comprehensively assess the actual impact of vibrations on structures and the surrounding environment. As many countries have incorporated frequency as part of the safety criteria for blast-induced vibrations, research on blast-induced vibration frequency has become particularly important. The frequency characteristics of blast-induced vibrations directly affect the propagation of vibration waves in the surrounding environment and structures, especially when the vibration frequency approaches the natural frequency of buildings, which may trigger resonance phenomena and further exacerbate damage to the structure. Furthermore, frequency analysis helps accurately assess the specific impact of vibrations in different frequency bands on the environment, particularly the differences in the effects of high-frequency and low-frequency vibrations on various types of structures. However, in existing studies, the discussion on blast-induced vibration frequency is somewhat fragmented and lacks systematic and in-depth analysis. Compared to PPV, the attenuation mechanism, influencing factors, and prediction methods for blast-induced vibration frequency have not yet formed a complete research framework. Frequency attenuation is affected by multiple factors [31], such as the explosion source to R, initiation method, geological conditions, and structural characteristics. However, existing studies are often limited to single or local factors, and most of these factors are difficult to quantify [28]. Moreover, while some empirical formulas and ML methods have achieved certain results in predicting blast-induced vibration frequencies, their general applicability and accuracy still need improvement.

To systematically investigate blast-induced vibration frequencies, this study first summarizes the safety criteria for blast-induced vibrations adopted in various countries. It then introduces characteristic frequencies in the spectral analysis of blast-induced vibrations and provides a comprehensive evaluation of related spectral analysis techniques. Subsequently, the study examines the attenuation laws and influencing factors of dominant frequencies, such as R, initiation methods, and borehole parameters. Finally, it explores frequency prediction methods based on empirical formulas and ML models, highlighting the advantages of ML techniques (e.g., ANN, SVM, and ANFIS) in capturing nonlinear features. The prediction results are compared using statistical metrics such as R², RMS, and MAE. This study summarizes the spectral analysis technique and compares its advantages and disadvantages in the analysis of blast-induced vibration signals, providing a reference for further research. By systematically analyzing the impact of various factors (such as R, initiation method, blasthole parameters, etc.) on blast-induced vibration frequency and combining the literature to explore the combined effects of these factors, this study provides a theoretical basis for optimizing the frequency prediction model, as well as scientific support for the design and safety evaluation of blasting plans. In-depth research on the characteristics, influencing factors, and prediction methods of blast-induced vibration frequency helps to accurately assess the impact of blast-induced vibrations on the surrounding environment and structures, providing scientific evidence and technical support for optimizing blasting design, technical methods and the sustainable development of blasting engineering.

2. Safety Criteria for Blast-Induced Vibrations

Assessing the impact of blast-induced vibrations on buildings and protected structures, as well as estimating their potential damage under vibration effects, is crucial for ensuring the safety of surrounding structures, personnel, and the environment. In the past, most countries commonly relied on single vibration parameters—such as displacement, velocity, or acceleration—as the primary criteria for evaluating blast-induced vibration intensity in safety regulations. However, practical experience has shown that a single parameter fails to accurately represent the relationship between vibration frequency and structural resonance. This limitation may, in certain cases, lead to either an underestimation or overestimation of the actual damage sustained by structures [19]. As data from engineering practices has accumulated over time, the limitations of single-parameter approaches have become increasingly evident. With the increasing accumulation of data from engineering practices, countries have gradually recognized the limitations of using a single parameter. As a result, vibration velocity and frequency metrics have been incorporated into safety evaluation standards to provide a more comprehensive and accurate framework for assessing blast-induced vibrations.

In the United States, blast-induced vibration safety criteria have been established for various industrial sectors based on the protection levels required for different structures. USBM and OSMRE developed their respective standards, which were later consolidated and refined to form the widely adopted international blast-induced vibration safety criteria. Figure 3 illustrates the key parameters and application scope of the combined USBM and OSMRE standards, providing valuable reference points for the global promotion of safety criteria in blasting engineering.

According to the regulations of the Office of Surface Mining, Reclamation, and Enforcement, the maximum PPV at any location outside the permitted area, including residential buildings, public buildings, schools, churches, or community or institutional buildings, shall not exceed the following limit (

Table 1. OSMRE (Office of Surface Mining, Reclamation and Enforcement, United States, Department of the Interior March, 1987) regulation.

Distance for Blasting Site (m)	Maximum Allowable Peak Particle Velocity (mm/s)
0–90	31.25
90–1500	25.00
1500 and above	18.75

).

Swiss standards (Figure 4) have also been developed for various building types, such as steel, reinforced concrete, brick–concrete, masonry, wooden attics, and historic and sensitive buildings. The corresponding PPVs are 30 mm/s, 18 mm/s, 12 mm/s, 8 mm/s, and 30–40 mm/s, 18–25 mm/s, 12–18 mm/s, 8–12 mm/s, for the frequency ranges 10–60 Hz and 60–90 Hz, respectively.

The German standard divides the building types into industrial, commercial, civil, and sensitive buildings, respectively, in the frequency ranges of 4–8 Hz, 8–30 Hz, and 30–100 Hz. The corresponding safe mass vibration combined velocities are 20 mm/s, 20–40 mm/s, 40–50 mm/s, 5 mm/s, 5–15 mm/s, 15–20 mm/s, 3 mm/s, 3–8 mm/s, and 8–10 mm/s (Figure 5).

The Indian standard classifies non-private buildings into residential buildings (brick and concrete), industrial buildings (crushed concrete and frame structures), and historically sensitive buildings, with three safe permissible vibration speeds within the frequency ranges of <8 Hz, 8–25 Hz, and >25 Hz with corresponding vibration velocities as given in Figure 6.

Chinese “Blasting Safety Regulations” (GB 6722-2014) [34] (Figure 7) suggest that the safety standards of blast-induced vibration are the ground building using PPV and frequency as a comprehensive criterion, with the PPV as f as the basis for correction reference. The structures for protection are classified as earth kiln caves, adobe houses, wool and stone houses, general civil buildings, industrial and commercial buildings, general ancient buildings, and monuments. In the three ranges of frequency <10 Hz, 10–50 Hz, and >50 Hz, the safety allowable PPVs corresponding to earthen kilns, adobe houses and woolen stone houses are 0.15–0.45 cm/s, 0.45–0.9 cm/s, and 0.9–1.5 cm/s, whereas those corresponding to general civil buildings are 1.5–2.0 cm/s, 2.0–2.5 cm/s, and 2.5–3.0 cm/s. The safety permissible vibration speed corresponding to industrial and commercial buildings is 2.5–3.5 cm/s, 3.5–4.5 cm/s, and 4.2–5.0 cm/s, while the safety permissible vibration speed corresponding to general ancient buildings and monuments is 0.1–0.2 cm/s, 0.2–0.3 cm/s, and 0.3–0.5 cm/s.

Based on relevant case studies,

Table 2. Relation between the vibration velocity and the security condition of the building.

Sources	Safety Condition of Buildings and Structures	Vibration Velocity (cm/s)
Railway Ministry Scientific Research Institute	Building safety	≤5
	Cracking and peeling of house wall plaster	12
	Large rocks rolling down steep slopes; small cracks appearing on the ground surface, causing damage to buildings	20
	Cracks appearing in soft, weak rocks; dry-stacked stone displacement; severe building damage	50
	Rock fractures, significant changes in terrain; complete building destruction	150
Institute of Geophysics	Slight damage to regular houses	10–15
	General damage to houses	>30
	Severe building damage, bedrock exposure causing cracks	≥60–70
Changsha Mining Research Institute	Formation of loose stones and small debris falling	8.1–11.1
	Formation of fine cracks or expansion of existing cracks	13.5–24.7
	Formation of 4~5 cm large cracks or expansion of existing cracks	45.8–81.5

summarizes the relationship between vibration velocity and building safety in certain cases in China. As shown in the table, when safety issues arise in buildings, vibration velocity does not fully reflect the building’s safety condition, and the existing safety standards have certain limitations in this regard. Therefore, relying solely on vibration velocity as an indicator for building safety evaluation may not fully capture the complexities in actual engineering situations. To better quantify engineering outcomes, it is recommended to combine more evaluation indicators, such as structural response, the building’s natural frequency, and other environmental factors, to more comprehensively and accurately assess the safety of buildings. By adopting a multi-dimensional evaluation approach, the accuracy of the assessment can be improved, providing stronger support for safety assurance in actual engineering projects.

With the widespread application of engineering blasting technology, the safety assessment and control of blast-induced vibrations have become critical to ensuring the safety of surrounding structures and personnel. Currently, many countries are gradually incorporating multiple parameters, such as vibration velocity and frequency, into their safety criteria. However, when vibration intensity exceeds a certain threshold, it can pose risks not only to the structural integrity and durability of the project itself but also to the stability of adjacent rock masses and buildings. Furthermore, blast-induced vibrations can have adverse physiological and psychological effects on individuals, such as stress, insomnia, and, in severe cases, psychological panic. To address these challenges, there is an urgent need to refine vibration assessment methods by integrating multi-parameter evaluation mechanisms to enhance prediction accuracy and reliability. By integrating ML models with high-precision vibration monitoring equipment, blasting operations can be dynamically adjusted, providing real-time feedback on vibration levels and adjusting parameters to ensure compliance with safety standards, thereby improving safety and prediction accuracy. Existing safety standards are typically based on approximate frequency ranges and vibration velocity, without precisely considering factors such as vibration sources, building natural frequencies, and vibration paths. As a result, these standards may not provide accurate assessments in certain specific environments. The impact of blasting vibration frequencies on buildings is complex; low-frequency vibrations can lead to resonant damage, while high-frequency vibrations near a building’s natural frequency may also cause damage. Therefore, relying solely on a single frequency range for safety assessment may not be sufficiently accurate.

3. Blast-Induced Vibration Frequency

3.1. Characteristic Frequencies in Blast-Induced Vibration Spectrum Analysis

When studying the spectral characteristics of blast-induced vibrations, the concept of characteristic frequency (i.e., the dominant frequency) is commonly used to describe these characteristics. The dominant frequency of blast-induced vibrations is generally categorized into three types: zero-cross dominant frequency, Fourier dominant frequency, and response spectrum dominant frequency. In his research on the effects of blast-induced vibrations under various detonation methods, Triviño [35] introduced the definition of the Centroid frequency to more comprehensively reflect the distribution characteristics of the vibration spectrum. In engineering applications, there is no absolute superiority of one type of dominant frequency over the others. Each type of dominant frequency has unique advantages in different scenarios [31]. For instance, the zero-cross dominant frequency is suitable for simplified analysis, the Fourier dominant frequency is better suited for spectral characteristic studies, and the response spectrum dominant frequency excels in dynamic response analysis. Therefore, when selecting the appropriate dominant frequency type, it is essential to consider the specific engineering requirements and the applicability of each frequency type, ensuring the selection of the most suitable frequency for the given practical context.

3.1.1. Zero-Cross Dominant Frequency

The zero-cross dominant frequency is determined based on the time coordinate of the zero-crossings in the time–history curve of blast-induced vibrations [36]. Typically, it can be directly calculated using Equation (1). Another common approach is to select the time interval corresponding to PPV as a half-cycle to compute the zero-cross dominant frequency, as illustrated in Figure 8a.

$$f_V=1/T_V$$

(1)

where f_V is zero-cross dominant frequency (Hz), T_V is the period (s).

The zero-cross dominant frequency is obtained through a preliminary estimation of the blast-induced vibration frequency, identifying the characteristic frequency encompassed by the velocity peak in the time–history curve of the vibration. This calculation method is simple and practical, capable of capturing the peak characteristic frequency, and is particularly suitable for vibration waveforms with distinct peaks. When the peak period of the vibration waveform is clearly identifiable, the zero-cross dominant frequency provides a more accurate representation of the frequency characteristics of blast-induced vibrations compared to the Fourier dominant frequency [37,38].

3.1.2. Fourier Dominant Frequency

The Fourier dominant frequency of blast-induced vibrations refers to the peak characteristic frequency within the vibration signal, which is calculated by applying a Fourier Transform to the time-history curve of the vibration [29]. In the velocity amplitude spectrum, the frequency corresponding to the maximum amplitude is defined as the dominant frequency. Fourier dominant frequency is typically the primary consideration during time-frequency analysis. Figure 8b illustrates the dominant frequency (f_d) derived from the Fourier Transform, with its calculation method as follows:

$$F(\omega )={\displaystyle {\int }_{-\infty }^{+\infty }v(t)e^{-j\omega t}dt}$$

(2)

where $F(\omega )$ is the Fourier amplitude spectrum (cm/s); $\omega$ is the angular velocity; $v(t)$ is the vibration time profile.

The advantage of the Fourier dominant frequency lies in its ability to accurately reveal the peak frequency while also presenting the range of the dominant frequency band. However, compared to the zero-cross dominant frequency, its higher computational complexity is a notable drawback.

3.1.3. Centroid Frequency

The centroid frequency is the average frequency of the velocity amplitude curve obtained by Fourier Transform of the blast-induced vibration time history curve. It can effectively characterize the signal average, also known as the average frequency. Calculate the average frequency (f_c) of the weighted average value using Equation (3).

$$f_{\mathrm{c}}={\displaystyle \frac{\sum f_{i}A(f_i)}{\sum A(f_i)}}$$

(3)

where f_c is the centroid frequency, f_i is the individual frequency in the Fourier amplitude spectrum, A(f_i) is the corresponding amplitude of the vibrational velocity spectrum.

To summarize the advantages and disadvantages of the three primary frequency types, refer to

Table 3. Comparison of the three types of dominant frequencies.

The Dominant Frequency	Advantages	Disadvantages
Zero-cross dominant frequency	1. Clearly reflects the frequency characteristics corresponding to the peak particle velocity. 2. Widely applied and recognized in engineering practice.	The accuracy of the calculation relies on the peak period of the vibration waveform. If the peak period is unclear or multiple peaks are present, it may lead to inaccurate results.
Fourier dominant frequency	1. Capable of reflecting the energy-concentrated regions of blast-induced vibration signals. 2. Facilitates the analysis of frequency components and energy distribution within the signal.	There is a limitation in that a single dominant frequency provides poor representativeness.
Centroid frequency	The distribution of the signal across the entire frequency range is considered, which to some extent mitigates the limitation of poor representativeness associated with a single dominant frequency.	1. The definition and calculation methods are relatively complex. 2. The specific applications and studies in blast-induced vibration analysis are relatively limited, and its advantages, disadvantages, and applicability require further exploration and validation.

. In vibration monitoring and assessment, it is crucial to choose an appropriate frequency analysis method based on specific operating conditions [35]. However, research on the relationship between blasting vibration frequencies and structural damage remains limited, particularly with regard to the selection of the appropriate main frequency type. There are still no clear guidelines for this choice. The selection of the main frequency is vital for an accurate evaluation of the impact of blasting vibrations on structures, especially when considering the risk of resonance. To address this, further investigation is urgently needed in the following areas: (1) expanding the theoretical analysis of blasting vibration frequencies to elucidate the mechanisms linking frequency and structural damage; (2) enhancing experimental validation to improve the precision of frequency monitoring; and (3) establishing standardized frequency analysis and evaluation protocols to guide engineering practice and provide a comprehensive assessment of the effects of blasting vibrations on structures.

3.2. Spectrum Analysis Techniques for Blast-Induced Vibrations

The Fourier Transform is a fundamental tool in spectrum analysis, enabling the conversion of time-domain signals into frequency-domain signals. By employing the Fourier Transform, signals (such as sound, images, etc.) can be decomposed into a superposition of sinusoidal waves (or complex exponential functions) of different frequencies, thereby facilitating the analysis of their spectral characteristics. Widely used in signal processing, communications, and image processing, the Fourier Transform provides an effective method for investigating the spectral characteristics of blast-induced vibration signals. For example, Geng et al. [39] utilized the Fourier Transform to study the spectral characteristics of blast-induced vibration signals. Their research revealed the energy distribution of signals across different frequencies and the time-varying behavior of vibration intensity, offering valuable references for structural safety assessments. Furthermore, Wang et al. [40] optimized the analytical approach by decomposing blast-induced vibration signals into various frequency ranges using Fourier decomposition. This allowed them to extract detailed signal characteristics and gain a more precise understanding of the nature and features of vibration signals. While Fourier analysis effectively captures the overall spectral information of a signal, it primarily focuses on the signal’s frequency-domain characteristics and cannot simultaneously reflect localized features in both time and frequency domains. For instance, it struggles to describe the frequency components present in a specific time interval, which is one of its key limitations. This challenge arises from the inherent trade-off between time resolution and frequency resolution in Fourier Transform, thereby constraining its application in the analysis of non-stationary signals.

With the advancement of digital signal processing technologies, the Short-Time Fourier Transform (STFT) was introduced to address the limitations of the standard Fourier Transform in capturing localized features in both time and frequency domains. By dividing the signal into small time segments for analysis, STFT can reflect the characteristics of the signal in both the time and frequency domains to a certain extent, enabling the extraction of frequency components over different time intervals. For instance, Wang [41] analyzed time–frequency distribution characteristics of vibration signals from open-pit bench blasting using STFT based on field-collected data at various blast distances. The study revealed that the energy of blast-induced vibration signals is primarily concentrated within the 30 Hz range, with a velocity direction pattern of vertical > horizontal radial > horizontal tangential. Additionally, the blasting energy exhibited an exponential decay with increasing R, with the near-field velocity attenuating more significantly than that in the far-field. As the elevation increased, the dominant frequency decreased, and the energy spectrum of far-field signals demonstrated multi-band and multi-peak characteristics. These findings provide important insights for evaluating the propagation patterns of blast-induced vibrations and structural responses. Despite the significant improvements offered by STFT in analyzing both time and frequency domains, it still has inherent limitations. The method provides a single resolution for time and frequency analysis, making it impossible to achieve both high time resolution and high-frequency resolution simultaneously. For example, in signals with rapidly changing frequency components, STFT struggles to accurately capture signal characteristics, thereby limiting its ability to perform detailed analysis of complex signals.

The concept of the Wavelet Transform was introduced in 1984 [42]. Building on the localization principles of the Fourier Transform, the Wavelet Transform overcomes the limitation of fixed time–frequency resolution and enriches the selection of wavelet bases, making it an ideal tool for the time–frequency analysis of non-stationary blast-induced vibration signals [43]. Yan et al. [44] proposed a fundamental framework for analyzing the energy distribution characteristics of blast-induced vibration signals based on the Wavelet Transform. By decomposing the blast-induced vibration signal into different frequency bands, the signal in each band remains a time-dependent signal. The time-variation patterns of vibration components within each frequency band are then analyzed to obtain the time–frequency energy distribution characteristics of the blast-induced vibration signal. This approach effectively reveals the variation patterns of blast-induced vibration signals in the time–frequency domain and provides significant insights for the analysis of complex vibration signal characteristics.

The Hilbert–Huang Transform (HHT) is a time–frequency analysis method that extracts instantaneous characteristics of signals based on local information. Compared to the Wavelet Transform, HHT demonstrates superior adaptability, making it particularly suitable for analyzing nonlinear and non-stationary signals. The HHT method primarily consists of two components: Empirical Mode Decomposition (EMD) and the Hilbert Transform. First proposed by Huang et al., this approach is regarded as a significant breakthrough in steady-state spectral analysis within the field of signal processing in recent years [45]. The prominent advantages of HHT lie in its adaptability and completeness, enabling a comprehensive representation of signal characteristics across the time, frequency, and energy domains. Liu et al. [46] employed HHT to analyze blast-induced vibration signals from a lead–zinc mine. By examining the signal characteristics from three perspectives—3D Hilbert spectra, marginal spectra, and instantaneous energy spectra—the study demonstrated that HHT effectively handles nonlinear and non-stationary signals, fully revealing their time–frequency characteristics while exhibiting notable advantages in signal reconstruction and precision. Dong et al. [47] utilized MATLAB to perform instantaneous phase optimization analysis based on HHT, improving the computation process of Hilbert spectra. The optimized HHT analysis enhanced the calculation accuracy of time–frequency and Hilbert spectra, making it more applicable for analyzing complex signal characteristics. Yuan et al. [48] applied the HHT method to analyze the time–frequency characteristics of recorded blast-induced vibration signals. Their research revealed that single-hole blast-induced vibration signals exhibit a concentrated distribution in the time domain but an uneven distribution in the frequency domain. Further analysis indicated that the dominant energy of blast-induced vibration signals is concentrated in several Intrinsic Mode Function (IMF) components corresponding to the primary frequency ranges of the vibrations. While HHT demonstrates significant advantages in time–frequency analysis, it heavily relies on the quality of EMD decomposition, which may be affected by data noise. Additionally, the computational complexity of HHT remains relatively high, necessitating further algorithmic optimizations to improve efficiency.

A summary of the four blast-induced vibration spectrum analysis techniques is presented in

Table 4. Comparison of four blast-induced vibration spectrum analysis techniques.

Spectrum Analysis Techniques for Blast-Induced Vibrations	Advantages	Disadvantages
Fourier Transform	1. Strong spectrum analysis capability, enabling a clear identification of the signal’s frequency components. 2. Exhibits linear properties, facilitating signal combination or decomposition. 3. The convolution theorem is highly useful in signal processing applications.	1. Loss of time-domain information, making it impossible to determine when specific frequency components occur. 2. Not suitable for analyzing non-stationary signals. 3. High computational complexity when processing large-scale data.
STFT	1. The fundamental algorithm is the Fourier Transform, which is straightforward to interpret in terms of its physical meaning. 2. Enables time–frequency analysis.	1. The window width is fixed and cannot be adaptively adjusted, resulting in the inability to simultaneously achieve optimal time and frequency resolution. 2. For non-stationary signals, when the signal undergoes rapid changes, the window function requires high time resolution. Conversely, during periods of relatively smooth waveform changes dominated by low-frequency signals, the window function requires high frequency resolution. STFT cannot accommodate both requirements simultaneously.
Wavelet Transform	1. Effectively addresses the limitations of the Fourier Transform and Short-Time Fourier Transform by providing a “time-frequency” window that adapts to changes in frequency. 2. Facilitates time–frequency analysis of non-stationary signals, offering information in both the time domain and frequency domain simultaneously.	The selection of wavelet bases is challenging, and different wavelet bases may yield varying analysis results.
HHT	1. Suitable for the analysis of nonlinear and non-stationary signals, making it better aligned with the characteristics of blast-induced vibration signals. 2. Capable of analyzing irregular signals and demonstrating strong adaptability to noise.	1. During the EMD decomposition process, mode mixing and over-decomposition may occur, which can compromise the accuracy of the analysis results. 2. The computational workload is significant, leading to relatively low efficiency.

. The Fourier Transform, as the foundation of spectrum analysis, has effectively supported the study of blast-induced vibration signals, but it is limited in its ability to achieve time–frequency localization. STFT alleviates some of the limitations of the traditional Fourier Transform, while the Wavelet Transform enhances the representation of time–frequency characteristics through multi-resolution analysis. However, the introduction of HHT offers enhanced adaptability and precision in handling nonlinear and non-stationary signals, allowing for a more comprehensive analysis of the instantaneous energy and frequency characteristics of blast-induced vibration signals. Moving forward, the analysis of blast-induced vibration signals should aim to integrate multiple time–frequency analysis methods, such as the Fourier Transform, Wavelet Transform, and HHT, to leverage their respective advantages and establish a comprehensive analytical framework. Additionally, incorporating ML techniques to develop intelligent algorithms for automated feature identification, combined with high-precision monitoring equipment to collect extensive data, will support the development of real-time monitoring and early warning systems.

Relying solely on spectral analysis may not provide a comprehensive assessment of vibration characteristics, especially when the vibration signals have complex time-domain features. Therefore, combining time-domain analysis with spectral analysis methods can offer a more complete evaluation. Time-domain analysis captures the instantaneous characteristics and abruptness of signals, while spectral analysis is better suited for describing the frequency characteristics of signals. Using both methods together can overcome the limitations of a single approach. The multi-scale analysis capability of wavelet transform has significant advantages in processing the local features of non-stationary signals. It can effectively analyze the short-term variation characteristics of blast-induced vibration signals, compensating for the limitations of traditional spectral analysis in time resolution. Therefore, wavelet transform can more accurately capture the nonlinear features and abrupt points of signals. The introduction of real-time monitoring technology can provide a dynamic blasting vibration evaluation pathway. Real-time feedback data helps engineers adjust blasting parameters in a timely manner based on changes in vibration signals. This feedback mechanism provides greater flexibility for vibration control, reducing the impact on surrounding environments and structures during blasting operations. Future research can further explore the integration of time-domain and frequency-domain analyses, using wavelet transform for more detailed signal analysis, and enhancing the accuracy and real-time nature of evaluations through real-time monitoring systems. Combining these technologies is expected to provide a more comprehensive and dynamic blasting vibration assessment method, thereby improving predictive capabilities and environmental protection levels in engineering practice.

4. Factors Influencing Blast-Induced Vibration Frequency

The influencing factors of blast-induced vibrations can be divided into blast-source factors and non-blast-source factors. Blast source factors include charge weight, charge structure, initiation sequence, delay time, minimum burden, and charge parameters. Non-blast source factors include R, free surfaces, topography, and propagation media. Charge weight is a key factor that determines explosive power and vibration intensity. As the charge weight increases, both explosive energy and vibration intensity increase. The charge structure affects the release of explosive energy and the propagation of vibration waves. In porous blasting, properly setting the delay time helps optimize energy distribution, reduces the superimposition effect of vibration waves, and thus lowers vibration intensity. Drilling depth and diameter also influence the propagation and release of explosive energy in the rock mass, thereby affecting the attenuation of vibration waves. Factors such as rock type, structure, and joint development also have a significant impact on the propagation and attenuation of blast-induced vibrations. Additionally, R is an important parameter for assessing the impact of blast-induced vibrations. As R increases, the amplitude and frequency of the vibration waves gradually attenuate. Therefore, by reasonably adjusting these blast source and non-blast source factors, the impact of blast-induced vibrations can be effectively controlled. For example, in a certain mining operation, optimizing the charge weight and delay time successfully reduced the impact of blast-induced vibrations on surrounding buildings and decreased the oversized block rate. After adjusting the blasting parameters, both vibration frequency and intensity remained within safe limits, effectively avoiding any impact on the residential areas surrounding the mining zone [49].

The response and damage of structures subjected to blast-induced vibrations are highly dependent on vibration frequency. Therefore, investigating the factors influencing blast-induced vibration frequency is not only crucial for engineering safety but also provides a foundation for mitigating environmental impact and minimizing potential harm to human activities and surrounding structures. Studies have shown that the dominant frequency of blast-induced vibrations is influenced by multiple factors, including blasting parameters, R, and rock properties [50].

Zhang [51], using a grey relational matrix analysis in matlab, identified the primary and secondary factors affecting blast-induced vibrations along with their correlation degrees. As shown in

Table 5. Degree of gray relational analysis [51].

${\mathbf{r}}_{\mathbf{ij}}$	Total Number of Holes	Total Charge Weight/t	Maximum Charge per Delay/kg	Maximum Charge of Single Holes/kg	Hole Depth/mm	Hole Diameter/mm	Charge Height/m	The Distance Between Front Hole/m	The Distance Between Holes/m	Row Spacing/m	Charge Structure
Vertical vibration speed V	0.76	0.85	0.83	0.52	0.78	0.69	0.77	0.77	0.50	0.51	0.77
Vertical vibration frequency f	0.75	0.79	0.79	0.51	0.76	0.75	0.77	0.76	0.51	0.51	0.78
V+f	1.51	1.63	1.62	1.03	1.54	1.44	1.54	1.53	1.01	1.02	1.55

, the parameters with a high correlation to blast-induced vibrations include the maximum charge per delay, total charge, charging method, charge length, hole depth, burden, the total number of blastholes, and hole diameter. Similarly, Zhao [52], through grey absolute correlation analysis, determined the influencing order of the dominant frequency as follows: blasting direction, elevation difference, distance, maximum charge per delay, number of segments, and total charge. This chapter provides a detailed discussion of the effects of various factors on the dominant frequency of blast-induced vibrations.

4.1. Charge Weight

Charge weight is a critical factor in determining the energy released during a blast, directly influencing both the intensity and frequency of the resulting vibrations. A larger charge weight releases more energy, typically leading to lower vibration frequencies. These low-frequency vibrations can significantly impact surrounding structures, particularly those that are more vulnerable, as such frequencies are more likely to induce resonance, potentially resulting in structural damage. Therefore, investigating the effect of charge weight on the dominant frequency of blast-induced vibrations is a fundamental step in controlling vibration frequency during blasting operations.

Sharpe [53] analyzed the spectral characteristics of blast-induced vibration signals with significantly different charge weights to investigate the influence of charge weight on vibration frequency. The study revealed that blast-induced vibrations with a smaller charge weight contained a higher proportion of high-frequency components, whereas those with a larger charge weight exhibited a lower dominant frequency. Similarly, Fan [54] compared the blast-induced vibrations generated by two different charge weights (110 kg and 10 kg). By applying the Fourier Transform to obtain the frequency spectrum, it was found that the dominant frequency for the larger charge weight was lower, approximately 20–30 Hz, while that for the smaller charge weight was higher, around 50–60 Hz. This finding confirmed the inverse relationship between charge weight and the dominant frequency of blast-induced vibrations. Zhong et al. [55], through an engineering case study, further demonstrated that when the blast source distance remained constant, an increase in the charge weight per delay led to a decreasing trend in the dominant frequency. This observation validated the negative correlation between charge weight and dominant frequency, as illustrated in Figure 9. Ling [56,57] applied wavelet packet analysis using matlab to process blast-induced vibration signals and obtained energy distribution variation curves for three types of vibration signals across different frequency bands (Figure 10). The study further analyzed the relationship between energy distribution and the maximum charge weight. The results indicated that as the charge weight increased, the proportion of high-frequency energy relative to the total energy decreased, causing the dominant frequency band of the blast-induced vibration signals to shift towards lower frequencies. Shi [58] developed vibration signal processing software in matlab based on the rearranged smoothed pseudo Wigner–Ville distribution (RSPWVD). By analyzing the time–frequency characteristics of vibration signals under different charge weights, it was observed that as the charge weight per delay increased, the dominant frequencies of the blast-induced vibration signals exhibited a decreasing trend and gradually stabilized at a constant value (Figure 11). Additionally, the duration of the first dominant frequency increased, eventually becoming the longest-lasting frequency. Hu [59] investigated the influence of frequency factors on blast-induced vibrations using real-world engineering cases. Based on field-measured data, the study established the relationship between dominant frequency and influencing factors. The analysis of charge weight per unit volume and dominant frequency data revealed that as the specific charge increased, the dominant frequency generally exhibited an increasing trend, as illustrated in Figure 12.

Proper distribution of explosives and charge structure is particularly critical. By accurately calculating the energy requirements of the blasting area and considering the attenuation characteristics of vibration waves during propagation, the explosive distribution plan can be optimized. This not only ensures efficient fragmentation but also significantly reduces the propagation range and intensity of vibrations. For example, rationally distributing the charge at different locations can reduce the vibration intensity at the blast source, or using staged charging techniques can control the release speed and method of explosive energy, thereby minimizing the impact of vibrations on the surrounding environment. To optimize explosive distribution, adopting modern delay blasting techniques is an effective approach. Millisecond delay blasting can effectively control the propagation and superposition effects of vibration waves, reducing vibration intensity and avoiding excessive instantaneous vibrations. Furthermore, appropriate borehole arrangement and charge structure (such as staged charging) can ensure efficient fragmentation while dispersing the energy release, thus reducing the concentrated impact on ground vibrations. In actual blasting operations, implementing real-time vibration monitoring is also an extremely effective strategy. By deploying a network of vibration sensors, engineers can monitor vibrations in real time and dynamically adjust parameters such as charge weight and delay time based on the monitoring data, ensuring that vibrations remain within acceptable limits.

4.2. The Distance from the Explosion Source to the Monitored Location (R)

Most empirical formulas for predicting blast-induced vibration frequency are derived using mathematical and statistical methods such as regression analysis, analysis of variance, and dimensional analysis. Studies have shown a strong correlation between vibration frequency and R. As R increases, the spatial extent of vibration disturbance expands, leading to a decrease in wavefront energy density and an increase in medium damping loss, which ultimately results in a reduction in vibration amplitude [60]. Typically, due to the high-frequency filtering characteristics of rock, the dominant frequency of blast-induced vibrations exhibits an exponential decay with R [61]. In recent years, researchers worldwide have reached a consensus on the relationship between blast-induced vibration frequency and R: as R increases, vibration amplitude decreases, and the dominant frequency shifts to lower values. This trend has been validated in multiple studies [36].

Aftabi [62] investigated the dominant frequency attenuation patterns of blast-induced vibrations in geomaterials with different natural frequencies (Figure 13). The study revealed a negative correlation between the dominant frequency and R, indicating that as R increases, the dominant frequency gradually decreases. At R = 1 m, the dominant frequency was highest in a granite environment, which exhibited a relatively high natural frequency. Even at R = 160 m, despite using the same explosive charge, the dominant frequency of granite-type rocks remained the highest. Furthermore, a significant difference in dominant frequency was observed between the nearest and farthest R. Chen [63] reported that as R increases, the lowest energy peak gradually decreases, the proportion of low- and mid-frequency energy diminishes, while the proportion of high-frequency energy increases. Additionally, the energy distribution becomes more dispersed, leading to a broader dominant vibration frequency bandwidth, with a tendency to shift towards higher frequencies. Li [64] noted that although the intensity of blast-induced seismic waves attenuates during propagation, the dominant frequency of the structural response tends to shift towards lower frequencies. Sun [65], using a physical similarity model, measured vibration data (as shown in Figure 14). The results indicate that the dominant frequencies in three different directions generally decrease with increasing R. Tian et al. [66] found that vibration energy rapidly attenuates with increasing R. By analyzing the variation patterns of power spectral density (PSD) at five measurement points (Figure 15), it was observed that in the vicinity of the blast source, energy is primarily concentrated in the low-frequency range. In the mid-range blast area, the energy distribution becomes broader, and the dominant frequency range expands. In the far-field blast zone, high-frequency energy gradually weakens, and the overall energy shifts toward the low-frequency range. Zhou et al. [67] utilized the LS-DYNA finite element software to analyze the attenuation patterns of blast-induced vibration frequencies for both spherical and cylindrical charge configurations (Figure 16). The study found that regardless of the charge shape, the dominant frequency of blast-induced vibrations does not follow a strictly monotonic decay with increasing blast source distance. Due to the multi-peak structure of the blast-induced seismic wave spectrum and the varying attenuation rates of high- and low-frequency components, local fluctuations or abrupt changes may occur during the attenuation process. Furthermore, Zhou [61] pointed out that the absorption characteristics of the medium contribute to frequency attenuation over distance. Additionally, during the attenuation of ground vibrations, rapid drops or fluctuations may occur. These fluctuations are typically observed in the mid-to-far field, and when a rapid drop occurs, the dominant frequency often decreases to approximately half of its original value (Figure 17). Shi [68] investigated the influence of R on the dominant frequency in three directions using field-measured data from two monitoring points (S1: R = 83 m; S2: R = 124 m). As shown in Figure 18, with increasing blast source distance, the proportion of low-frequency energy in the signal increases, and the dominant frequency of blast-induced vibrations tends to shift toward the lower frequency range. Tian [69] examined the spectral characteristics of seismic waves generated by blasting excavation for foundation pits. As illustrated in Figure 19, blast-induced seismic waves exhibit a broad frequency range, which gradually narrows as R increases. When R becomes sufficiently large, this variation becomes less pronounced. At R = 40 m, the dominant frequency is 37 Hz, while at R = 60 m, it decreases to 25 Hz. With increasing R, the dominant frequency decreases, and the dominant frequency band shifts towards lower frequencies while becoming narrower.

Multiple studies consistently indicate that the dominant frequency of blast-induced vibrations exhibits a decreasing trend with increasing R, with low-frequency components becoming predominant, particularly in the far-field region. Specifically, as R increases, the vibration amplitude decreases, and the dominant frequency gradually shifts to lower values. Moreover, the spectral characteristics of blast-induced vibrations undergo significant changes with increasing R, as low-frequency energy progressively increases while high-frequency components attenuate. Additionally, factors such as geomaterial properties, charge configuration, and medium absorption characteristics influence the frequency attenuation process to varying degrees, playing a critical role in the evolution of vibration frequency.

4.3. Detonation Method

Delay blasting has a direct impact on rock fragmentation, muck pile formation, and ground vibrations [70], making it an effective technique for enhancing rock breakage while reducing blast-induced vibrations. It not only improves rock fragmentation efficiency but also influences the impact characteristics of blast-induced vibrations, thereby affecting the dominant frequency of vibration [71].

Shi [72] observed in an open-pit mine that the dominant frequency increases with increasing delay time. Zhao [73], while keeping R and charge weight constant, analyzed the time–frequency characteristics of four different delay times (25 ms, 42 ms, 65 ms, and 100 ms). Each delay time was tested four times, and the range of dominant frequency variations across the trials is shown in Figure 20. The results indicate that different delay times correspond to different dominant frequencies of blast-induced vibrations. The dominant frequency initially increases with increasing delay time, then decreases. Therefore, to effectively control blast-induced vibrations and protect nearby structures, it is essential to optimize the selection of delay time. Lou [74], through field experiments, conducted a comparative study on the influence of different initiation delays (t_m) using Ordinary detonators and Orica detonators. The relationship between detonator type, delay time, and dominant frequency is illustrated in Figure 21. The results indicate that for Ordinary detonators, the occurrence of the highest dominant frequency in the blast zone was more dispersed, with lower data stability. In contrast, for Orica detonators, the highest dominant frequency occurred at more concentrated time points, demonstrating higher data stability. Additionally, in both blast zones, the dominant frequency of vibration waves exhibited a trend of initially decreasing and then increasing as the delay time increased. This suggests that delay time has a significant impact on the dominant frequency of blast-induced vibrations. Proper selection of delay time can optimize vibration effects, thereby mitigating potential damage to buildings and surrounding equipment. Ma [75] conducted a delayed blasting model experiment to mitigate the vibration effects of vertical shaft blasting. Using two rings of parallel boreholes as a prototype, the study implemented three different initiation delays (0 ms, 25 ms, 50 ms). As shown in Figure 22, the delayed blasting amplitude spectrum exhibited a greater number of peak fluctuations, a higher dominant frequency, and a more complex vibration frequency composition. Additionally, the vibration waveform of delayed blasting was more intricate than that of simultaneous initiation. The amplitude of the frequency spectrum in delayed blasting was significantly lower compared to simultaneous blasting, indicating that delayed initiation effectively reduces blast-induced vibration intensity.

The initiation method influences the propagation of stress waves and the transmission of explosive energy in borehole blasting, thereby affecting rock fragmentation efficiency. Different initiation methods exhibit variations in blast-induced vibration frequency [35].

Peng [76], through experimental vibration frequency analysis, found that compared to row-by-row initiation, hole-by-hole initiation resulted in an overall upward shift in vibration frequency, which is beneficial for protecting nearby structures. Additionally, in row-by-row delayed initiation using conventional detonating tubes, the maximum peak frequency remained below 20 Hz, aligning closely with the natural frequency of most buildings. This confirms that different initiation methods lead to distinct variations in dominant vibration frequency, and by adjusting the blasting technique, vibration mitigation can be effectively achieved. Zhang [77], based on theoretical analysis and wavelet analysis of five signals using matlab, determined the energy distribution across different frequency bands (Figure 23). The frequency range of simultaneous blasting was relatively narrow and concentrated in the lower-frequency range, typically below 10 Hz. In contrast, delayed blasting exhibited a broader frequency range: row-by-row delayed initiation generally remained below 30 Hz, while hole-by-hole initiation extended up to 50 Hz. The increase in the number of delay intervals from simultaneous blasting to row-by-row and further to hole-by-hole initiation caused vibration energy to concentrate in the mid-to-high frequency range, leading to an increase in the dominant frequency. Adjusting the borehole delay time directly influences the interaction of explosive waves between boreholes, thereby reducing blast-induced vibrations and improving overall blasting performance.

Duan [78], using field experiments, conducted hole-by-hole initiation while keeping the borehole layout parameters constant. The study analyzed blast-induced vibrations under odd–even sequential initiation and wave-like sequential initiation. Figure 24 illustrates the radial dominant frequency of blast-induced vibrations at different blast source distances for various initiation sequences, while Figure 24 shows the numerical sequence of initiation timing. A comparison of different initiation methods reveals variations in the dominant frequency. At R = 30 m and R = 60 m, the radial dominant frequency was highest for wave-like sequential initiation, whereas at R = 90 m, the highest radial dominant frequency was observed for hole-by-hole sequential initiation. These findings confirm that the initiation sequence significantly influences the dominant frequency of blast-induced vibrations. Li et al. [79], by studying the impact of different detonation methods on the dominant frequency of blast-induced vibrations, found that switching from the original V-shaped detonation to surface delay sectional detonation significantly altered the frequency of the blast-induced vibrations. As shown in Figure 25, delayed detonation, compared to V-shaped detonation, results in a significant increase in the dominant frequency.

Before optimizing the delay time, engineers should carefully assess the geological conditions at the work site, including factors such as rock hardness, density, elastic modulus, and layering structure. These factors directly affect the propagation speed and attenuation characteristics of the blast wave, which in turn influences the intensity and range of vibrations. By using numerical simulation methods to simulate vibration propagation under different delay times, engineers can theoretically predict the impact of delay time on vibration amplitude and propagation distance and identify the optimal delay time setting. The simulation process should account for the effects of different geological conditions on vibration propagation to ensure that the optimization plan is adaptable under various operational conditions. Due to the complexity and uncertainty of blasting operations, theoretical models alone may not fully predict the optimal delay time. Therefore, engineers can conduct small-scale experiments on-site, adjusting the delay time incrementally and observing the changes in blast effects and actual vibration performance. Based on experimental results, the delay time can be further adjusted to achieve the best blasting effect and minimize environmental damage. In practical operations, the optimization of delay time not only needs to consider vibration intensity but also the safety of the protected targets. For example, the relationship between the natural frequency of structures and the dominant frequency of vibrations generated by blasting must be considered. Delay time adjustments should avoid making the vibration frequency approach the resonance frequency of the structure, thereby minimizing damage to buildings or equipment. Real-time monitoring technologies, such as vibration sensors and data acquisition systems, can track vibration intensity and frequency during the blasting process, providing real-time feedback to optimize delay time adjustments. This can provide valuable experience for subsequent blasting operations and help engineers quickly adjust delay times to reduce vibration damage.

4.4. Borehole Parameters

In blasting engineering, borehole diameter rarely changes due to drill bit size limitations. Numerous studies have shown that borehole diameter affects the performance of explosives and influences the intensity of blast-induced vibrations [80]. Additionally, the velocity of detonation (VoD) of explosives tends to increase with increasing borehole diameter [81].

Liang et al. [82] selected two datasets with minimal differences in blast source distance and PPV to compare the dominant frequencies of blast-induced vibrations in three directions for different borehole diameters. As shown in Figure 26, the dominant frequency of vibrations in small-diameter boreholes is higher than that in large-diameter boreholes. Similarly, Xu et al. [83], based on field measurements, demonstrated in Figure 27 that boreholes with a diameter (HD) of 90 mm exhibit a higher proportion of energy in the low-frequency range compared to boreholes with HD = 76 mm. Furthermore, as R increases, the low-frequency components tend to increase to varying degrees. Under similar R conditions, the dominant frequency of 90 mm boreholes is lower than that of 76 mm boreholes, confirming that small-diameter boreholes are more effective in reducing blast-induced vibrations compared to large-diameter boreholes.

Borehole diameter has a significant impact on the dominant frequency of blast-induced vibrations. Smaller boreholes typically generate higher dominant frequencies, whereas larger boreholes result in lower dominant frequencies. Additionally, small-diameter boreholes exhibit a higher proportion of energy in the low-frequency range and are more effective in controlling blast-induced vibrations compared to large-diameter boreholes. As R increases, low-frequency components gradually become more dominant, and smaller boreholes prove to be more effective in regulating vibration frequency. However, research on the precise mechanisms governing the relationship between borehole diameter and blast-induced vibration frequency remains limited. Future studies should further investigate the influence of borehole diameter on blast-induced vibrations under various geological conditions, charge structures, and explosive types. Such research will contribute to the development of optimized vibration control strategies, enhancing both safety and efficiency in blasting engineering.

4.5. Free Surface

In rock blasting, the detonation of an earlier explosive charge creates a new free face for the subsequent delayed charge [84]. In blasting operations, free faces facilitate rock fragmentation and allow fractured rock to displace into surrounding media. Additionally, free faces influence wave reflection and refraction [85]. The presence of free faces significantly impacts blast-induced vibrations [86], improving blasting efficiency while reducing vibration velocity and energy, thereby altering dominant frequency characteristics. As the number of free faces increases, explosive consumption decreases, and the propagation characteristics of stress waves change accordingly.

Wu [87] investigated the energy distribution characteristics of blast-induced seismic waves under different numbers of free faces using real-world engineering cases (Figure 28). The study found that when the number of free faces increased from one to two or from two to three, the percentage of energy distributed within the 62.5–125 Hz frequency band (Band 5) slightly increased, while the energy distribution in the 0–31.25 Hz frequency band (Band 4) decreased. This indicates that an increase in free faces shifts energy towards higher frequencies. Such a shift is beneficial in avoiding the natural frequency of buildings and structures, making the strategic use of free faces more effective for blast vibration safety. Yang et al. [88], based on experimental and numerical simulation studies, pointed out that when the compressive stress wave generated by an explosive load reaches a free face, it reflects as a tensile wave. The interaction between the reflected and original stress waves shortens the rise time and duration of loading in distant areas, increasing the load frequency. As a result, in the presence of free faces, the dominant vibration frequency increases, and the proportion of high-frequency energy in the total energy rises. Moreover, the closer the blast source is to the free face, the higher the blast-induced vibration frequency. From a vibration frequency perspective, well-designed free faces can help reduce structural damage from blast vibrations. Chi [89], utilizing AOK time–frequency distribution and wavelet analysis, examined blast-induced seismic wave characteristics by analyzing PPV, dominant frequency, frequency band energy distribution, and energy duration for one, two, three, and four free faces. As shown in Figure 29, as the number of free faces increases, the dominant frequency band widens, and the dominant frequency increases.

The role of free faces in rock blasting has a significant impact on blast-induced vibration characteristics. By altering the propagation and reflection of stress waves, free faces facilitate rock fragmentation while reducing explosive consumption. Studies have shown that an increase in the number of free faces shifts energy towards higher frequencies, thereby mitigating low-frequency vibrations that could otherwise affect nearby buildings and lead to resonance-induced damage. Additionally, an increase in free faces raises the vibration frequency and broadens the dominant frequency band, which helps to reduce structural damage. In practical applications, the strategic design of free face quantity, type, and positioning can enhance blasting efficiency while minimizing environmental vibration impacts. The shape, type, and size of free faces also have a significant influence on blasting performance and ground vibrations. A greater number of free faces results in more complex stress wave reflection and tensile interactions, thereby enhancing rock breakage efficiency. The size and shape of the free face determine the looseness and ejection direction of fragmented rock, affecting the minimum resistance line for explosive charges. Although research on the effects of free face type and size on blast vibrations remains limited, studies indicate that different types of free faces (e.g., air-filled, water-filled, or unloaded) and the characteristics of the adjacent rock surface significantly impact blast-induced vibrations. Therefore, in engineering practice, it is crucial to fully consider the influence of free faces. Through the optimal design of free face quantity and positioning, blasting efficiency can be improved while vibration impact on the surrounding environment is reduced, ensuring safety and effectiveness in blasting operations.

4.6. Burden

The burden is defined as the minimum distance between the explosive charge center and the free face. It directly influences the explosive consumption per meter of the borehole, fragmentation distribution, and fragment dispersion effect. The rock mass surrounding different burden lengths experiences varying degrees of damage effects, leading to distinct failure patterns. Variations in burden length result in different rock breakage mechanisms, which subsequently affect the surrounding rock mass, vibration intensity, and vibration frequency.

Uysal [90] investigated the effects of different burden lengths on blast-induced vibrations under two working conditions. Figure 30 presents the dominant frequency curves for different burden lengths at R = 100 m under a specific condition. The results indicate that burden length has a significant impact on vibration characteristics, though no clear trend is observed. Shi [58] found that under similar blasting conditions, increasing the front-row burden leads to a significant reduction in the dominant frequency of vibration waves, while the duration of the dominant frequency gradually increases. The Pearson correlation coefficient between burden length and dominant frequency was calculated as -0.58, indicating a negative correlation (Figure 31), meaning that as the burden increases, the dominant frequency decreases. Li [91], using similarity theory, constructed a concrete model and conducted single-hole blasting tests with varying burden sizes. Figure 32 illustrates the variation in dominant frequency in the radial direction under different specific charge weights and burden lengths. The results show that as the specific charge weight increases, the dominant frequency also increases. For the same specific charge weight, a larger burden results in a higher dominant frequency. Zhang [92], employing similarity principles, conducted blasting experiments on a scaled concrete bench model. The study revealed that the dominant frequency of blast-induced vibrations gradually decreases with an increase in the minimum burden (B) (Figure 33).

Burden length has a significant impact on the dominant frequency of blast-induced vibrations, although research findings indicate that its influence does not follow a completely consistent pattern. Generally, increasing the burden length tends to reduce the dominant frequency of vibration waves, while the duration of the dominant frequency gradually increases. Additionally, a larger burden is associated with a higher dominant frequency, and as the specific charge weight increases, the dominant frequency also increases. While existing studies have revealed certain trends, the precise mechanisms governing the relationship between burden length and blast-induced vibration frequency require further investigation.

4.7. Explosive Type

Different types of explosives exhibit distinct blasting characteristics. Leidig [93] conducted experimental studies on ground vibrations induced by different types of explosives. The results showed that black powder has a longer explosion duration, leading to the formation of long fractures, which in turn enhance low-frequency ground vibrations. In contrast, Explosive B does not generate long fractures due to the inability of its detonation gases to drive extensive crack propagation, resulting in higher-frequency vibrations.

Xue [94], using the Hilbert–Huang Transform (HHT), obtained the marginal energy spectrum of blast-induced vibration signals. The findings revealed that single-base propellant generates lower vibration energy levels compared to emulsion explosives. Additionally, single-base propellant vibration signals are primarily concentrated in the 0–5 Hz and 15–25 Hz frequency ranges, whereas emulsion explosives exhibit a more dispersed energy distribution across 0–50 Hz. Furthermore, as R increases, the energy in the 0–5 Hz range gradually decreases, while the energy becomes more concentrated in the 15–25 Hz range. Chai [95], analyzing the energy release mechanisms of explosives within rock formations, combined real-time blast-induced vibration monitoring data to compare the vibration characteristics of two explosives (GIFT and KUBELA). As shown in Figure 34, with increasing specific charge weight, the dominant frequency of blast-induced vibrations for both explosives initially increases and then decreases, though with noticeable differences between the two. This confirms that the performance characteristics of different explosives significantly influence the dominant frequency of blast-induced vibrations.

Different types of explosives have a significant impact on the dominant frequency of blast-induced vibrations. Studies have shown that black powder, due to its longer explosion duration and extensive crack formation, enhances low-frequency energy, whereas other explosives, such as Explosive B, fail to drive long fractures, resulting in a higher proportion of high-frequency energy. The energy distribution of vibration signals also varies between single-base propellant and emulsion explosives. The single-base propellant primarily concentrates its energy in the 0~5 Hz and 15~25 Hz frequency ranges, whereas emulsion explosives exhibit a more dispersed energy distribution across 0~50 Hz. As R increases, the low-frequency energy of single-base propellants gradually diminishes, shifting toward higher frequency bands. Moreover, the performance characteristics of different explosives influence the dominant frequency of blast-induced vibrations, and with increasing specific charge weight, the dominant frequency initially increases and then decreases.

4.8. Geological Condition

The propagation of seismic waves in soils and rocks is influenced by the physical properties of the medium, including density, elastic modulus, and Poisson’s ratio. These characteristics determine the wave propagation speed, attenuation rate, and reflection/refraction behavior. As the propagation distance increases, the dominant frequency of the vibration waves gradually decreases, showing a negative correlation. This is because high-frequency components decay faster than low-frequency components during propagation. The attenuation of vibration frequency is mainly caused by factors such as medium absorption, scattering, and waveform transformation. The absorption characteristics of different soils and rocks vary significantly, affecting the rate of frequency attenuation. In geotechnical materials with different natural frequencies, the dominant frequency attenuation patterns of vibration waves are diverse and may be related to the resonance effects of the medium. When the frequency of the vibration wave is close to the natural frequency of the soil or rock, resonance occurs, leading to a significant increase in amplitude, which may result in structural damage. The resonance effect is one of the key factors controlling the amplification of vibration frequency, especially more pronounced in weak or fractured geotechnical media.

Different types of rock masses and terrains can influence wave propagation, including characteristics such as wave velocity, amplitude, and frequency, thereby affecting blast-induced vibrations [29]. Álvarez-Vigil [31] utilized artificial neural networks to predict PPV and the frequency of open-pit mine blasts. Based on the physical properties of the rock mass, explosive characteristics, and blasting design, the study predicted surface PPV and frequency, concluding that different rock types influence the variation of dominant frequency in blast-induced vibrations. Tripathy [96] combined explosion data from similar rock types to develop attenuation equations for basalt, granite, quartzite, and sandstone. The study found that the frequency of blast-induced vibrations is significantly influenced by the rock medium, terrain, and geological conditions. Ma [97] employed nonlinear structural dynamic analysis to study structural damage caused by underground accidental explosions. The results indicated that different terrains and propagation media affect the generation and transmission of dominant frequency in blast-induced vibrations. Surface vibrations in soil have a lower dominant frequency compared to those on rock surfaces, and as the soil layer depth increases, the dominant frequency of vibrations becomes lower.

Properties such as the hardness, density, and elastic modulus of rocks, as well as the damping effects of the medium, all influence the characteristics of vibration propagation. Zhang [98] applied wavelet transform and response spectrum methods to study the frequency band characteristics of blast-induced vibrations in jointed rock masses and high-gradient slopes. The study concluded that in rock masses with developed joints, the PPV and energy of each frequency band attenuate, and the dominant frequency band of the vibration signals is high and wide. Song [99] conducted multiple field tests to quantify the characteristics of the propagation medium using basic rock mass indicators and integrity coefficients. The results showed that the dominant frequency band range of the signal waveform, after wavelet packet decomposition, increases as the integrity coefficient of the propagation medium decreases.

Geostress has a significant impact on the distribution of blast-induced vibration energy and frequency. Yang [100] pointed out that as geostress increases, the proportion of low-frequency energy in blast-induced vibrations also increases. As shown in Figure 35, with increasing geostress, the energy of blast-induced vibrations becomes increasingly concentrated in both low and high frequencies. When geostress exceeds 50 MPa, the low-frequency vibration energy becomes roughly equal to the high-frequency vibration energy. This suggests that at geostress levels above 50 MPa, the vibration energy of the surrounding rock is primarily influenced by strain energy rather than solely by the vibration energy induced by the explosive load. Yang [101] conducted a study that indicated, at different levels of geostress, the amplitude–frequency spectrum of coupled vibrations, as shown in Figure 36. As the in situ stress P0 increases, the amplitude significantly increases in the low-frequency region, while the amplitude only slightly increases in the high-frequency region. This suggests that the in situ stress level primarily affects the amplitude of low-frequency vibrations.

Using data fitting methods and signal processing tools, Liu et al. [102] conducted a comparative analysis of vibration intensity, frequency, and power under conditions with and without elevation differences. The study also examined the negative impact of negative elevation differences on blast-induced vibration mechanisms. The results indicate that negative elevation differences accelerate the attenuation of PPV, dominant frequency, and energy in blast-induced vibrations. Under negative elevation conditions, blast vibrations are primarily concentrated in the low-frequency range of 10~30 Hz, with vertical vibration energy experiencing the fastest attenuation rate. Additionally, the dominant frequency exhibits a negative exponential decay with increasing R. The accelerated frequency attenuation caused by negative elevation differences is mainly attributed to the propagation of seismic waves in concave terrain, which increases the vertical travel distance of wave transmission. Zhang et al. [103], based on open-pit mine blast vibration monitoring data, analyzed the dominant frequency characteristics of blast-induced vibrations at different elevations. Figure 37 illustrates the vertical dominant frequency under varying elevation differences, showing that as elevation increases, the dominant frequency tends to shift toward lower frequencies.

The type of rock mass, terrain conditions, in situ stress, and other geological factors have a significant impact on the dominant frequency and energy distribution of blast-induced vibrations. The physical properties of different rock masses and terrain media influence wave propagation characteristics, including wave velocity, amplitude, and frequency. For example, vibration frequencies differ between soil surfaces and rock surfaces, with deeper soil layers generally exhibiting lower dominant frequencies. In situ stress also plays a crucial role in vibration characteristics. As stress levels increase, the energy distribution between low- and high-frequency vibrations tends to balance. Notably, at stress levels exceeding 50 MPa, rock mass vibration energy is primarily governed by strain energy rather than blast-induced loading energy. Additionally, negative elevation differences accelerate the attenuation of vibration frequencies, particularly in the low-frequency range. In such cases, vertical vibration energy experiences the fastest attenuation rate. Underground cavities amplify low-frequency components, increasing the risk of resonance. In rock masses with developed joints and fractures, the frequency components of the blast seismic waves are more complex, and the relative energy of the blast vibrations is mainly distributed across a wider frequency band, which is caused by the multi-mode response of the fractured rock mass structure. Additionally, the hardness of the propagation medium significantly affects vibration propagation: when the propagation medium is harder, the vibration amplitude is lower, the dominant frequency is higher, and the duration is shorter; while in softer media, the vibration frequency is lower. The denser the rock and the smaller its porosity, the faster the propagation speed of stress waves, with less energy loss, resulting in a higher dominant frequency. Therefore, these geological factors significantly affect the characteristics of vibration propagation and dominant frequency attenuation and should be fully considered in blasting safety assessments.

This chapter covers a range of factors that influence the dominant frequency, but these factors are often discussed independently. However, in blasting engineering, these factors are not isolated; they interact and influence each other in complex ways. For example, geological conditions are key media for wave propagation and have a significant impact on the dominant frequency of blast-induced vibrations. As described in reference [61], different rock types lead to significant variations in the dominant frequency of blast-induced vibrations, depending on R. Similarly, reference [80] indicates that variations in borehole diameter and blasting distance also cause significant differences in dominant frequency. Additionally, parameters such as charge weight and delay time affect the release and propagation of explosive energy, which in turn influences the dominant frequency. Therefore, a comprehensive analysis of blast-induced vibrations must consider the interactions between these factors.

Due to the limited data in the literature, this study, under the same conditions, uses four quantifiable influencing factors (borehole diameter, R, maximum charge per delay, and total charge weight) as sub-sequences, with the dominant frequency as the main sequence, to calculate the correlation between each factor and the dominant frequency. To eliminate the influence of dimensional factors, each indicator was normalized before calculation. According to Equations (4)–(7), it was found that all four influencing factors are correlated with the blast-induced vibration dominant frequency. The correlation results are shown in

Table 6. Correlation degree calculation results.

Borehole Diameter	R	Maximum Charge per Delay	Charge Weight
0.73	0.54	0.65	0.8

. Among them, charge weight has the highest correlation with the dominant frequency, followed by borehole spacing and maximum charge per delay. The sensitivity ranking of each indicator is as follows: charge weight > borehole spacing > maximum charge per delay > R.

$${\mathsf{\Delta }}_{ij}=\left|x_{ij}^{\prime }-y_{ij}^{\prime }\right|$$

(4)

$${\mathsf{\Delta }}_{\max }=\max ({\mathsf{\Delta }}_{ij})$$

(5)

$${\mathsf{\Delta }}_{\min }=\min ({\mathsf{\Delta }}_{ij})$$

(6)

Then, the correlation coefficient of X_i with Y at point k can be expressed as:

$${\gamma }_{ij}(k)={\displaystyle \frac{{\mathsf{\Delta }}_{\min }+\eta {\mathsf{\Delta }}_{\max }}{{\mathsf{\Delta }}_{ij}+\eta {\mathsf{\Delta }}_{\max }}}$$

(7)

where η is the distinguishing coefficient, typically taken as 0.5.

The correlation degree of X_i with Y can be expressed as:

$${\gamma }_{ij}={\displaystyle \frac{1}{n}}\sum _{k=1}^n{\gamma }_{ij}(k)$$

(8)

In the above formulas, X_i represents the influencing factors, and Y is the evaluation index, namely the slope stability coefficient. Using Equations (4)–(7), the correlation degree of each factor with the stability coefficient can be obtained. The correlation degree falls within the range [0, 1], where a value closer to 1 indicates a stronger relationship between the factor and the stability coefficient, meaning the factor is more sensitive to slope stability. Conversely, a lower correlation degree signifies less sensitivity.

By optimizing key parameters such as charge weight, charge structure, initiation sequence, and delay time, the propagation distance and duration of vibration waves can be effectively controlled, significantly reducing interference to the surrounding environment while ensuring that the vibration frequency remains within safe limits, thus safeguarding the safety of infrastructure such as underground pipelines and buildings and minimizing the risk of potential damage. To reduce noise pollution, blasting operations are often accompanied by significant noise, which impacts the quality of life for surrounding residents. By optimizing the initiation method and adjusting charge weight, the noise levels produced by the explosion can be reduced, thus protecting residents’ mental health and mitigating the negative impact on the ecological environment. In terms of reducing structural damage, through precise calculations and analysis, the appropriate charge weight and charge structure can be chosen to ensure that explosive energy is evenly distributed within the target area. These optimization measures effectively reduce the impact and damage of the explosion on surrounding buildings, preserving the structural integrity of the buildings and lowering the risk of structural damage caused by blasting operations.

Borehole parameters (such as borehole diameter, depth, and spacing) have a significant impact on the dominant frequency of blast-induced vibrations. Borehole parameters directly influence the propagation mode and energy distribution of the blast wave. Different borehole diameters and depths result in different energy release patterns, thereby affecting the vibration frequency. In addition, the presence of free surfaces affects the propagation path and reflection characteristics of the blast wave, which plays an important role in controlling vibration frequency. Although previous studies have indicated that the number and arrangement of free surfaces influence vibration frequency, the specific mechanisms of these effects have not been fully theoretically explained. Similarly, charge structure and delay time are key factors that determine the rate of explosive energy release and the propagation characteristics of vibration waves. Proper delay time settings can optimize energy distribution, reduce the superimposition effect of waves, and thus lower frequency peaks. Current research tends to focus on the impact of delay time on vibration intensity, but the specific effects of delay time on frequency characteristics still require further systematic study. Moreover, factors such as minimum burden and charge structure also affect the vibration frequency generated during the blasting process to some extent. Changes in the minimum burden can alter the propagation path and propagation speed of the blast wave, thereby affecting the frequency components of the vibration. By appropriately selecting the charge structure (e.g., continuous or staggered charging) and setting suitable charge parameters, the dominant frequency can be effectively adjusted, optimizing the blasting design and reducing the risk of structural damage. More importantly, current research lacks an in-depth exploration of the specific effects of the uncoupling coefficient, blasthole density coefficient, and explosive unit consumption on the dominant frequency. These parameters are significant in controlling energy distribution and vibration frequency during blasting. Further study of these factors will help establish more precise frequency prediction models. The mechanisms of these factors’ influences are not only related to frequency prediction but may also provide new perspectives for multi-factor joint assessment.

5. Prediction of Blast-Induced Vibration Dominant Frequency

5.1. Empirical Equation

The dominant frequency of blast-induced vibrations is influenced by a complex, nonlinear relationship with various factors, and there is currently no quantitative prediction system. Scholars typically use empirical formulas, ML algorithms, and numerical simulations to predict the dominant frequency of blast-induced vibrations. The earliest vibration frequency prediction formula was proposed by Savage [104] in 1966, which only considered R, neglecting other factors that could influence frequency. Later, Jiao [105] introduced a new formula in 1995, which incorporated shear wave velocity and charge quantity, thereby expanding the original prediction formula. Tang [106], building on previous research, further proposed an empirical formula for predicting the dominant frequency. However, these empirical formulas based on experimental data have certain limitations in their applicability, as they fail to comprehensively account for all potential influencing factors, such as geological conditions and detonation methods.

Building on previous research, Zhang [107] derived a relationship curve between dominant frequency, charge quantity (Q), and R through experimental studies, as shown in Figure 38. The study found that when R is fixed, the dominant frequency decreases as Q increases. Conversely, when Q is fixed, the dominant frequency decreases as R increases. Furthermore, when the charge ratio remains constant, the dominant frequency decreases with an increase in R. Gao et al. [108], referencing the Sadowski formula, proposed a prediction equation for blast-induced vibration frequency. Based on the least squares fitting of charge ratio and frequency ratio, the fitted curve is shown in Figure 39. As can be seen from the figure, with the increase in the proportional dosage, the proportional frequency shows a negative correlation. Meng [109] conducted a regression analysis of measured blast-induced vibration data from the Shenzhen Meishi Power Plant and proposed a functional relationship between blast-induced vibration frequency and the main influencing factors (charge quantity and R). The study revealed the variation patterns of the dominant frequency with R and charge quantity in bench-blasting seismic waves, with a linear fitting curve shown in Figure 40. A prediction formula for dominant frequency under specific blasting conditions was derived. Based on the linear fit, the R² value reaches 0.95, demonstrating a strong correlation and reflecting the accuracy of the predictive formula. Lu [110] derived a decay formula for the dominant frequency of blast-induced vibrations from spherical explosive charges based on the analytical solution of elastic waves excited by dynamic internal pressure in spherical cavities and the attenuation law of blast-induced vibration frequency. Wang [110] determined the seismic wave amplitude and frequency for different R, Q, and rock properties. Peng [111], through dimensional analysis, introduced an elevation difference coefficient and derived a decay formula for blast-induced vibration dominant frequency considering elevation effects. Aldas [37] combined the decay formula of the dominant frequency with the PPV decay equation and proposed a PPV-based dominant frequency prediction formula. A summary of various blast-induced vibration dominant frequency prediction formulas is presented in

Table 7. List of equations.

Scholars	Equations	Description of the Equation
Savage [104]	$f={\displaystyle \frac{1}{k{\log }_{10}R}}$	f is the dominant vibration frequency (Hz); Q is the total charge (kg); R is the distance from the explosion source to the monitored location (m); a, a2 are constants to be determined; Kf is the frequency coefficient; Cs is the transverse wave velocity of the rock; K(k), α, β, bs, as are coefficients related to topographic and geologic conditions; PPV is peak particle velocity (cm/s); Cp is the longitudinal wave velocity in the rock mass; H is the difference in elevation from the centre of the explosion source to the measurement point (cm); d is the diameter of the borehole. ρ is the rock density.
Jiao [105]	$f={\displaystyle \frac{K_f{C}_s^{\frac{7}{5}}}{Q^{\frac{1}{3}}}}{({\displaystyle \frac{Q^{\frac{1}{3}}}{R}})}^{{\displaystyle \frac{2}{5}}}$
Tang [106]	$f=K{({\displaystyle \frac{Q^{\frac{1}{3}}}{\lg R}})}^{{\displaystyle \frac{1}{2}}}$
Zhang [107]	$f={\displaystyle \frac{K}{R}}{({\displaystyle \frac{Q^{\frac{1}{3}}}{R}})}^R$
Gao [108]	$f=K{\displaystyle \frac{PPV}{R}}{({\displaystyle \frac{Q^{\frac{1}{3}}}{R}})}^R$
Meng [109]	$f={\displaystyle \frac{1/R}{a(\sqrt[3]{Q}/R)+a_{2}R}}$
Lu [110]	$f=\xi {\displaystyle \frac{C_p}{{\alpha }^{1/2}}}{({\displaystyle \frac{Q^{1/3}}{R}})}^{\beta }$
Wang [112]	$f=b_s\cdot r^{a_s}\cdot f_{\mathrm{q}}$
Peng [111]	$f\cdot R=k{({\displaystyle \frac{Q^{1/3}}{R}})}^{\alpha }{({\displaystyle \frac{H}{R}})}^{\beta }$
Aldas [37]	$f={\displaystyle \frac{1}{\alpha R}}\ln ({\displaystyle \frac{R^{\beta -1/2}}{k{Q}^{\beta /2}}})$

. When predicting the dominant frequency, it is essential to strictly adhere to relevant industrial norms and safety standards to ensure the safety of the surrounding environment. Most existing dominant frequency prediction formulas are based on Q and R, with factors such as geological conditions, elevation differences, and wave velocity introduced through theoretical analysis and mathematical methods. However, in actual blasting safety assessment standards, typically, only frequency and vibration velocity are considered without fully accounting for the combined impact of multiple factors. Further exploration of the relationship between dominant frequency and other influencing factors and the incorporation of multi-factor assessments into safety standards, will provide more comprehensive safety guarantees for blasting operations.

The derivation and application of basic equations are of significant importance when evaluating blast-induced vibrations. These equations typically involve multiple parameters, such as Q, R, elevation differences, the physical properties of rocks, and coefficients related to topography and geological conditions. In the derivation of these equations, scholars usually rely on theoretical analysis, experimental data, and statistical analysis to gradually build mathematical models that describe the characteristics of blast-induced vibrations. By considering multiple influencing factors, prediction formulas for the dominant frequency of blast-induced vibrations can be established, allowing for more accurate predictions of the vibration frequencies caused by explosions. Through the analysis of these formulas, the influence of each parameter on the vibration characteristics can be understood, which aids in optimizing blasting designs, reducing the impact of vibrations on the surrounding environment and structures, and providing a scientific basis for assessing the safety of surrounding environments and structures affected by blast-induced vibrations.

Most of the empirical equations proposed by existing scholars do not explicitly define the frequency segment thresholds, and thus, the compatibility between these formulas and safety standards requires further investigation. In contrast, DIN 4150 provides a direct basis for selecting the dominant frequency type (such as zero-cross dominant frequency or Fourier dominant frequency) in engineering practice by dividing the frequency into different ranges (e.g., 4–8 Hz, 8–30 Hz, etc.). Furthermore, the frequency weighting method in the ISO standard offers theoretical support for multi-parameter safety assessments, especially for the joint control of PPV and dominant frequency. Therefore, future research can explore how to better integrate existing empirical equations with these international safety standards to improve the accuracy and practicality of predictions. Despite significant progress in the field of blast-induced vibration prediction, several challenges remain. Traditional dominant frequency prediction methods are mainly based on the Sadove formula, which, although considering factors such as R and charge quantity, faces limitations due to the substantial differences between these factors and the actual engineering environments (such as open-pit mining or hydropower blasting). These discrepancies affect the prediction accuracy and applicability. With the advancement of society and improvements in living standards, the demand for higher prediction accuracy has become stricter. As a result, traditional prediction methods have increasingly shown their limitations, especially with the development of testing instruments that allow for the collection of more influencing factors, which gradually weakens the effectiveness of traditional empirical formulas. For open-pit blasting or complex geological conditions, dimensional analysis can effectively improve the Sadowski formula by incorporating factors such as elevation differences and rock mass tensile strength, thereby enhancing predictive capabilities. However, there remains a lack of effective methods for handling certain hard-to-quantify influencing factors or discontinuous variables (such as the number of faults), which limits the application of dimensional analysis in complex blasting scenarios.

5.2. Artificial Intelligence (AI) Techniques for Blast-Induced Vibration Dominant Frequency

Due to the numerous blasting parameters involved, deriving the relationship between complex parameters and the dominant frequency of blast-induced vibrations is a challenging task. In recent years, ML, as a significant research direction in the field of artificial intelligence, has attracted widespread attention from scholars. Many researchers have utilized ML algorithms and techniques to predict the dominant frequency of blast-induced vibrations.

5.2.1. Artificial Neural Network (ANN)

Artificial Neural Network (ANN) is mechanisms that learn the complex relationships between inputs and outputs by adjusting the connection weights between neurons. A common structure of the ANN architecture is shown in Figure 41 [113].

Recent years have seen significant advantages of Artificial Neural Network (ANN) in predicting the dominant frequency of blast-induced vibrations. Álvarez-Vigil et al. [31] considered various parameters influencing blast-induced vibrations, including rock mass characteristics, explosive properties, and blasting design parameters and developed an ANN model to predict blast-induced vibration. By training the ANN model, they predicted PPV and dominant frequency. The prediction results of this model showed a high degree of correlation compared to traditional methods. The correlation coefficients between the predicted and measured PPV and dominant frequency were 0.98 and 0.95, respectively (Figure 42). These results, which approach 1, demonstrate the effectiveness and accuracy of the method in predicting blast-induced vibrations.

In 2006, Khandelwal et al. [114] proposed a model that uses neural network methods to predict PPV and dominant frequency. The study considered various factors influencing ground vibrations, including rock properties, blasting parameters, and explosive types and developed a neural network model consisting of an input layer, hidden layer, and output layer. A comparison between the neural network prediction model, the multiple regression analysis prediction model, and measured values, as shown in Figure 43, demonstrated that the proposed neural network model provides higher accuracy in predicting PPV and dominant frequency compared to multiple regression analysis. This method offers an important reference for mine blasting design and vibration control. Later, in 2009, Khandelwal et al. [115] introduced an improved three-layer feedforward backpropagation neural network model. This model was trained using 154 sets of experimental and monitoring data, with the input layer containing 10 parameters related to blasting design and rock properties, and the output layer predicting PPV and dominant frequency. A comparative analysis of the correlation between the ANN prediction results and the calculated values from Equation (9) (shown in Figure 44) revealed that the ANN model offers higher accuracy and reliability in predicting PPV and dominant frequency, further validating the application value of neural networks in blast-induced vibration prediction.

$$\begin{array}{l}f=23.5183-0.0004[hd]-4.3022[B]+4.4027[S]-0.0027[Q_{\max }]-0.0015[R]\\ -1.32[BI]+6.3759[E]+0.002[v]-0.1399[Pv]-0.8346[VOD]\end{array}$$

(9)

where hd is the hole depth (m), B is burden (m), S is the spacing (m), Q_max is the maximum charge per delay (kg), R is the distance from the explosion source to the monitored location (m), BI is the geotechnical index, E is the Young’s modulus (GPa), v is the Poisson’s ratio (m/s), P_v is the P-wave velocity (m/s), VOD is the velocity of detonation of explosive (km/s).

Zhang et al. [116] developed a BP (backpropagation) neural network model for predicting PPV and dominant frequency, using five key factors as the primary influencing variables: total charge weight, maximum charge per delay, number of blasting segments, R, and site conditions. They used empirical formulas as a comparison method to predict PPV and dominant frequency. The results, shown in Figure 45, demonstrate that the BP neural network model has significant advantages in prediction accuracy. Specifically, the relative errors in the BP neural network model’s predictions of PPV and dominant frequency are 12.50% and 13.59%, respectively, while the relative errors for the predictions made using traditional empirical formulas are 26.18% and 11.16%. These data indicate that the BP neural network model exhibits significantly higher accuracy in predicting PPV and also shows greater reliability in predicting dominant frequency, further validating the effectiveness and superiority of the BP neural network model.

Singh et al. [117] developed an artificial neural network model for predicting PPV and dominant frequency by considering blasting parameters and the physical and mechanical properties of rock as input parameters. The correlation coefficient between the predicted and measured dominant frequency values is 0.8369 The results of the study indicate that this prediction method offers high accuracy and provides important reference information for vibration control and safety assessment in blasting operations.

Görgülü et al. [118] selected three different open-pit mines and conducted multiple measurements. Using R, maximum charge per delay, and the distance between holes as input parameters, they developed an artificial neural network model to predict dominant frequency. The results showed that the predictions made using the developed neural network model exhibited a high correlation coefficient (R² = 0.83~0.98). Compared to regression analysis models, the artificial neural network model demonstrated greater applicability and accuracy in estimating and evaluating the intensity of blast-induced ground vibrations.

Artificial Neural Networks (ANN) have achieved significant success in the field of blast vibration prediction. By integrating various influencing factors, ANN can accurately predict PPV and dominant frequency, demonstrating advantages in high-precision prediction and strong adaptability. Compared to traditional methods, ANN shows a particularly evident advantage in handling complex nonlinear relationships and multivariable influences. However, the training process of ANN is complex and requires substantial data and computational resources. Furthermore, ANN is prone to becoming trapped in local optima, leading to instability in prediction results, which limits its widespread application in certain scenarios. Although previous studies have considered the impact of variations in the propagation medium, the digital description of geological conditions remains rudimentary, which restricts the predictive performance of the ANN model. Future research should explore more effective methods for quantifying geological factors to enhance the prediction capability and applicability of ANN models in different engineering environments.

5.2.2. Support Vector Machine (SVM)

Support Vector Machine (SVM) is a powerful ML algorithm widely used for classification and regression problems. Figure 46 illustrates the core concept of SVM. The primary objective of SVM is to find an optimal hyperplane that maximizes the boundary between data points of different classes, thus enabling efficient data separation. SVM can handle not only linear data but also nonlinear data by introducing kernel functions. Moreover, it performs effectively in high-dimensional spaces [119].

Wang et al. [120] used RBF neural networks, BP neural networks, and SVM algorithms to predict PPV and dominant frequency, comparing the results with the traditional Sadove formula. The study showed that the prediction accuracy of these three algorithms was superior to that of traditional methods. However, with a limited sample size, both BP and RBF neural networks exhibited certain limitations in predicting PPV and the dominant frequency of blast-induced vibrations. In contrast, the SVM algorithm outperformed the neural networks in terms of prediction accuracy. A comparison between the measured values and the predictions from each method is shown in Figure 47. In predicting dominant frequency, the relative errors for RBF and BP neural networks were relatively large (Figure 48), and the overall prediction performance was poor. The main reason for this is the limited number of engineering samples, which affected the stability of the prediction models. On the other hand, SVM maintained relative errors mostly within 10% when predicting dominant frequency, demonstrating higher prediction accuracy. This result indicates that SVM can effectively control errors within a reasonable range when predicting blast-induced vibration characteristic parameters, showcasing strong overall prediction capability, particularly in small sample data applications.

Support Vector Machine (SVM) has proven highly effective in addressing nonlinear problems and small sample data, particularly in complex vibration prediction tasks, owing to its strong generalization ability. However, the training process of SVM is intricate and computationally demanding, requiring careful parameter selection. Moreover, its efficiency diminishes when dealing with large-scale datasets. In this study, while SVM demonstrates effective prediction of blast vibration frequencies, the training efficiency declines substantially as the dataset size increases, which in turn affects both the timeliness and accuracy of the predictions.

5.2.3. Adaptive Neuro-Fuzzy Inference System (ANFIS)

Adaptive Neuro-Fuzzy Inference System (ANFIS) is a hybrid intelligent algorithm that combines neural networks and fuzzy inference [121]. It integrates the reasoning capability of fuzzy logic with the learning ability of neural networks, forming a system that is nonlinear, adaptive, and capable of learning. ANFIS can autonomously adjust fuzzy rules and membership function parameters through training with input–output data, thereby improving the model’s prediction accuracy and generalization ability. To illustrate the working process of ANFIS, let X and Y represent the inputs and f represent the output. The fuzzy inference process and rules are shown in Figure 49.

Singh et al. [20] applied the ANFIS model to predict blast vibration characteristic parameters, considering various multidimensional factors such as blast design parameters, rock physical properties, and explosive performance, and specifically predicting the dominant frequency of blast vibrations. The research results showed that the ANFIS model performed excellently in predicting the dominant frequency. After prediction, the correlation coefficient between the model’s predicted values and the measured data reached as high as 0.9988, fully validating its prediction accuracy and reliability. A comparison of the predicted and measured dominant frequency values further confirmed the high accuracy of the ANFIS model in predicting blast vibration characteristics. ANFIS demonstrated multiple advantages in blast vibration prediction, including nonlinear modeling capabilities, adaptive learning mechanisms, flexible fuzzy reasoning, and high-precision predictions, making it an effective tool for predicting vibration characteristic parameters in complex blasting environments.

ANFIS integrates the learning capabilities of neural networks with the inferential power of fuzzy logic, making it particularly effective in dealing with uncertainty and imprecision, especially when the data are incomplete or inaccurate. By dynamically adjusting fuzzy rules, ANFIS can adapt to varying data distributions, demonstrating considerable flexibility. When applied to nonlinear and complex systems, ANFIS provides high prediction accuracy. However, its training process, which involves the optimization of both fuzzy rules and neural network parameters, comes with high computational complexity. This becomes especially problematic in large-scale data processing, potentially leading to excessive consumption of computational resources. Moreover, due to the dependence on both fuzzy reasoning and neural network learning, the training process of ANFIS is time-intensive and sensitive to initial conditions. Additionally, ANFIS requires high-quality input data, and the presence of noise or missing data can significantly impair its predictive performance.

5.2.4. Gene Expression Programming (GEP)

Gene Expression Programming (GEP) is a novel adaptive evolutionary algorithm inspired by the structure and function of biological genes. GEP is developed based on Genetic Algorithms (GA) and Genetic Programming (GP), combining the advantages of both while overcoming their shortcomings [122]. A significant feature of GEP is its simple and efficient encoding method, which allows it to solve complex problems while maintaining strong versatility and adaptability.

Wang et al. [123] proposed an optimized GEP algorithm by combining Principal Component Analysis (PCA) with GEP and applied it to predict PPV and dominant frequency. The algorithm uses PCA to preprocess the input data, eliminating correlations between inputs, and constructs a BV-PCA-GEP prediction model. The experimental results show that the BV-PCA-GEP algorithm predicts PPV and dominant frequency with errors of 8.51% and 6.47%, respectively, which are significantly better than the 12.14% and 12.92% errors obtained by the BP neural network algorithm. The predictions made by the BV-PCA-GEP model are closer to the measured values, demonstrating higher prediction accuracy and stability compared to other algorithms like BP neural networks. Additionally, Dindarloo’s [124] research further validated the superior performance of Gene Expression Programming (GEP) in predicting dominant frequency. The study found that GEP’s MAPE and R² were 4.7% and 0.97, respectively, while the corresponding values for ANN were 9.3% and 0.81. As shown in Figure 50, GEP outperforms ANN in terms of prediction accuracy.

Gene Expression Programming (GEP) is an adaptive evolutionary model based on evolutionary algorithms that effectively address complex nonlinear problems, exhibiting strong global search capabilities. Compared to traditional genetic algorithms and genetic programming, GEP uses a simpler encoding method, which allows for more efficient problem-solving and the ability to flexibly adapt to the unique characteristics of different problems. However, the training process of GEP may face challenges such as slow convergence or non-convergence, particularly when dealing with complex data characteristics, necessitating additional fine-tuning. While GEP can automatically generate highly adaptable models, the expressions it produces can be complex, lacking interpretability and making it difficult to derive intuitive insights from the model. In this study, while GEP optimized the prediction model for blast vibration frequency using evolutionary algorithms, the complexity of the dataset resulted in longer training times, and the instability of the training process may undermine the model’s reliability. Furthermore, the lack of interpretability in the models generated by GEP poses a potential limitation for practical engineering applications.

5.2.5. Random Forest (RF)

Random Forest (RF) is a powerful ensemble learning method that makes predictions by combining multiple decision tree models, offering high accuracy and robustness. A common structure of the RF architecture is shown in Figure 51. During the training process, Random Forest generates random subsets from the training data using bootstrap sampling, and at each node, it randomly selects a subset of features for splitting. This randomness effectively reduces the risk of overfitting in individual decision trees. Finally, Random Forest combines the predictions from all trees through voting or averaging, resulting in stronger generalization capability. It is widely used in classification and regression tasks [125].

Dong et al. [126] selected nine factors, including total charge weight, horizontal distance, elevation difference, and burden, as influencing variables, and used the developed SVM and RF models to predict the dominant frequency of blast-induced vibrations at a copper mine in China. The results show that both the SVM and RF models had small prediction errors compared to the measured values (as shown in Figure 52). Among them, the RF model performed particularly well in blast-induced vibration prediction, further validating its accuracy and applicability in this field. By combining multiple decision tree models, Random Forest effectively reduces the overfitting risk of individual decision trees, improving both the stability and accuracy of the predictions.

5.2.6. Optimization Algorithm

With the rapid development of ML, optimization algorithms have shown significant advantages in predicting blast-induced vibration frequencies. These algorithms can handle complex nonlinear relationships, adapt to the variations in different blasting scenarios, and possess global optimization capabilities. When combined with computer and automation technologies, they enable intelligent and automated prediction processes, significantly improving prediction accuracy and efficiency. This provides more reliable data support for blasting engineering.

Yue et al. [127] used the PSO algorithm to search for the optimal combination of the key parameters γ and σ in the LSSVM model, with seven indicators—total charge weight, maximum charge per delay, elevation difference, burden, powder factor, stemming length, and R—serving as the input for the LSSVM model. Additionally, blast PPV and dominant frequency were used as output variables, resulting in the development of a PSO-LSSVM prediction model, as shown in Figure 53. The prediction performance of the PSO-LSSVM model was significantly better than that of the BP neural network and LSSVM models, with results closer to the actual values, as shown in Figure 54. The comparison revealed that the BP neural network exhibited the lowest prediction accuracy and the highest volatility, with average relative errors of 42.37% and 42.05%, and root mean square errors (RMSE) of 13.39% and 13.29%, respectively. The unoptimized LSSVM model showed average relative errors of 15.57% and 18.74%, and RMSE of 4.92% and 5.92%, representing a significant improvement in prediction accuracy over the BP neural network. The PSO-LSSVM model exhibited the smallest fluctuation range in relative errors, with average relative errors of 3.31% and 6.38%, and RMSE of 0.98% and 2.02%, demonstrating higher prediction accuracy and stability.

Zhou et al. [128] combined rough set theory with fuzzy neural networks (FNN) to accurately predict PPV and the dominant frequency of blast-induced vibrations. By applying rough set theory to eliminate false data and simplify secondary attributes, they obtained the optimal attribute set. Based on the RS-FNN model (Figure 55), they selected parameters such as maximum charge per delay, R, propagation medium conditions, total charge weight, elevation difference, and delay time as input data, with PPV and dominant frequency as decision attributes for prediction analysis. The experimental results (Figure 56) showed that the predicted values were in close agreement with the measured values, with an average error of 7.2% for PPV and 3.9% for dominant frequency. These results were significantly better than the 19.1% error from the Sadowski empirical formula, demonstrating that the RS-FNN model offers high prediction accuracy and meets practical engineering requirements.

In summary, the machine learning methods for predicting dominant frequency are summarized in

Table 8. Machine learning summary.

Researcher	Model	Input Parameters	State-of-the-Art
Álvarez-Vigil [31]	ANN	RMR, blast-control point relative arrangement, R, borehole diameter, stemming, S, B, instantaneous charge, total charge weight, No. of blastholes, explosive detonation velocity	MRE = 2.77; R2 = 0.9012
Khandelwal [114]	ANN	Hole depth, B, S, maximum charge per delay, R, geotechnical (compressive strength/tensile strength), E, v, Pv, VOD	MAE = 0.24467; R2 = 0.9086
Khandelwal [115]	ANN	Hole diameter, average hole depth, average burden, average spacing, average charge length, average explosive per hole, R, blastability index, E, v, Pv, velocity of detonation of explosive, density of explosive	MAPE = 6.99; R2 = 0.9868
Zhang [116]	ANN	Total charge weight, maximum charge per delay, number of blast segments, R, characteristics of site conditions	The relative errors were 11.16% and 13.59% for comparisons with S and Tang, respectively
Singh [117]	ANN	Borehole diameter, No. of holes, average hole depth, B, S, average top stemming, average explosive per hole, R, blastibility index (ratio of compressive to tensile strength), E	R2 = 0.8369
Görgülü [118]	ANN	R, explosive amount per delay, number of drill holes, distance between holes, burden, stemming, resistivity, P-wave and S wave velocities	R2 = 0.83~0.98
Wang [120]	RBF	R, maximum charge per delay	Average relative error is 11.08%
	ANN		Average relative error is 12.58%
	SVM		Average relative error is 6.72%
Singh [20]	ANFIS	(B/S), (explosive/ delay), R, E, Pv, VOD	R2 = 0.9988
Wang [123]	ANN	Total charge weight, maximum charge per delay, elevation difference, minimum burden, R, delay time, rock integrity coefficient, rock solidity coefficient, angle between measurement point and minimum burden direction	Average relative error is 12.92%
	GEP		Average relative error is 7.00%
Dindarloo [124]	ANN	Borehole diameter, No. of holes, hole depth, B, S, stemming, maximum charge per delay, R, radial R	MAPR = 9.3%; R2 = 0.81
	GEP		MAPR = 4.7%; R2 = 0.97
Dong [126]	SVM	maximum charge per delay, Total charge weight, horizontal distance, R, B, presplit penetration ratio, integrity coefficient, angel of minimum resistance line tomeasured point, VOD	Average relative error is 8.07%
	RF		Average relative error is 12.04%

. Training an accurate ML prediction model indeed requires a certain amount of historical data. The specific data requirements depend on the model’s complexity, the diversity of input parameters, and the precision of the target prediction. In general, the prediction accuracy of a model is closely related to the quality and quantity of training data. For more complex models, especially deep learning models, a large amount of historical data is usually needed to effectively capture underlying patterns. However, in some cases, particularly for newly initiated projects or regions with complex geological conditions, historical blasting data may be scarce, which can limit the direct application of ML models. To address the issue of data scarcity, strategies such as data augmentation, simulated data generation, and transfer learning can be employed. Data augmentation can simulate data by perturbing or expanding existing data, simulated data generation can be carried out through physical modeling, and transfer learning can improve model prediction capabilities by utilizing data from related fields. This way, even in the absence of sufficient local data, good model training can still be achieved. Regarding computational resources, ML model training and prediction processes do require a certain level of computing power. As model complexity increases, the computational demand also rises. In industrial environments, this typically means the need for high-performance computing equipment or reliance on cloud computing services for large-scale data processing and real-time predictions. While computational cost is an important consideration, with the continuous development of computing technology and the reduction in hardware costs, more and more ML models are now capable of real-time applications in industrial environments. Furthermore, by optimizing model structures and algorithms, such as using more efficient training methods, simplifying models, and reducing unnecessary computational load, computational complexity can be further reduced, improving real-time performance and making ML methods more adaptable to industrial field demands.

In the field of blasting vibration prediction, both traditional empirical formulas and ML methods have their advantages and application scenarios. For example, the Sadowski empirical formula, which is typically based on simple mathematical relationships, only considers a limited number of influencing factors. These formulas are easy to use, fast in calculation, and suitable for simple scenarios with limited data. However, they fail to fully account for the nonlinear characteristics of blasting vibrations and cannot consider multiple vibration parameters (such as PPV and dominant frequency), resulting in poor accuracy and applicability in complex blasting environments. BP neural networks can handle multiple influencing factors and simultaneously predict both PPV and frequency. This aligns with the requirements in many national standards to consider both vibration velocity and frequency. Although BP neural networks are computationally resource-intensive and have slower training speeds, they can provide accurate and comprehensive prediction results when the data set is large. In cases with fewer samples, SVM typically offers higher prediction accuracy. SVMs are effective in handling complex nonlinear problems and reducing noise interference found in traditional formulas. Although the training process for SVM is more complex and requires the selection of an appropriate kernel function, it outperforms BP neural networks and RBF neural networks when the sample size is insufficient. ANFIS combines the advantages of fuzzy logic and neural networks, enabling it to handle complex nonlinear problems. It has strong nonlinear modeling capabilities and an adaptive learning mechanism, which results in higher accuracy in blasting vibration predictions, particularly in complex environments. However, the training process for ANFIS is relatively complex. GEP has powerful global search and optimization capabilities and exhibits high adaptability and prediction accuracy when handling complex nonlinear problems. It can address complex issues that traditional empirical formulas cannot solve. Although its training process is also complicated, GEP holds high value in complex engineering applications. In practical engineering applications, selecting the appropriate prediction method is crucial. By combining traditional empirical formulas with ML algorithms, the accuracy and adaptability of blasting vibration predictions can be effectively improved to meet the demands of complex blasting environments.

A comparison of empirical formulas and ML methods for predicting the dominant frequency of blast vibrations is presented in

Table 9. Comparison of prediction methods.

Prediction Method	Advantages	Disadvantages	Comparison of Individual Cases Under the Same Working Conditions
Empirical equation	1. Simple to calculate and cost-effective to implement. 2. Capable of providing reasonable predictions for relatively simple blast vibration scenarios. 3. Requires minimal data, making it suitable for situations with small sample sizes.	1. Difficult to address complex nonlinear relationships and the impact of multiple variables. 2. Sensitive to variations in geological conditions, blasting parameters, and other factors, limiting its adaptability to complex environments. 3. Prediction accuracy is relatively low and significantly decreases as the complexity of the conditions increases.	Condition 1 [115]: R2 = 0.098 Condition 2 [116]: Average relative error = 11.16% Condition 3 [118]: R2 = 0.17
AI Techniques	1. High accuracy, with the ability to automatically learn complex patterns from large datasets. 2. Capable of adapting to various geological and blasting conditions, with strong modeling capabilities for handling multivariable influences. 3. Exhibits strong adaptability to large and complex datasets.	1. The training process is complex and demands significant data and computational resources. 2. The model requires high-quality data, as noise or missing data can negatively impact its performance. 3. The model is prone to overfitting and necessitates the use of appropriate regularization techniques to improve its generalization capability.	Condition 1 [115]: R2 = 0.9086 Condition 2 [116]: Average relative error = 13.59% Condition 3 [118]: R2 = 0.83~0.98

. In blast vibration prediction studies, traditional methods are typically effective in areas with relatively simple and stable geological conditions, such as uniform rock layers. In the absence of historical data, traditional methods have an advantage, as empirical formulas generally do not require large datasets for model training. Additionally, when the required prediction accuracy is relatively low, traditional empirical formulas can offer reasonable vibration predictions, especially in scenarios where the blasting conditions are simple and controllable. By comparing the prediction accuracy of ML and traditional empirical formulas, it was found that ML significantly improves prediction accuracy. For example, combining the aforementioned literature, when using different methods for predicting the dominant frequency under the same working conditions, the correlation coefficient between the multiple regression analysis formula prediction and the measured data is only 0.098, whereas the correlation coefficient between the ANN prediction and the measured data is as high as 0.9086. In another case, the results derived from the established artificial neural network model showed a high correlation coefficient (R² = 0.83~0.98), compared to the regression analysis model (R² = 0.17), demonstrating greater applicability and accuracy. However, issues such as insufficient sample size and poor generalization ability of ML may lead to lower prediction accuracy. For example, in one case, the average relative errors of the BP neural network prediction and the formula prediction were 13.59% and 11.16%, respectively, showing a decrease in prediction accuracy with ML compared to empirical formulas. The natural frequencies of different geological conditions vary (e.g., granite 12.4 Hz, rhyodacite 9.5 Hz, magnetite 7.4 Hz), and the dominant frequency of blast-induced vibrations also differs under various geological conditions. Therefore, in different environments, the prediction of the dominant frequency generally shows variation. Furthermore, the influence of geological factors on the dominant frequency has also been validated. Research has found that with an increase in tectonic stress, the proportion of low-frequency energy in blast-induced vibrations increases. As a result, researchers often use geological conditions as input parameters for prediction models to improve accuracy. This not only validates the importance of geological conditions in predicting the dominant frequency but also emphasizes the necessity of considering geological diversity.

Empirical equations, due to their simplicity, ease of use, quick implementation, and widespread application, have been commonly adopted in many practical engineering projects. However, these equations also have certain limitations. They are typically based on empirical data under specific conditions, have a limited scope of applicability, and lack accuracy in complex scenarios. Moreover, empirical equations generally assume linear relationships, which makes it difficult to address the complex nonlinear relationships between blast-induced vibrations and multiple factors, further limiting their prediction accuracy and reliability. Therefore, when using empirical equations, it is crucial to fully consider their scope of application and limitations and to make appropriate adjustments and corrections based on the specific situation. In contrast, ML demonstrates advantages in blast vibration prediction, such as high accuracy, strong adaptability, and the ability to analyze multiple factors simultaneously. It can handle complex data and nonlinear relationships, adapting to different geological conditions and blasting parameters. However, this approach requires high-quality and sufficient data; if the data are insufficient or inaccurate, the prediction results may be affected. Additionally, the training process for ML models is computationally complex, requiring more computational resources and time. Furthermore, some ML models exhibit the “black-box effect”, meaning the prediction results lack transparency, making it difficult to interpret the model’s decision-making process. Therefore, when applying ML for blast vibration prediction, factors such as data acquisition, computational resources, and model interpretability need to be considered comprehensively. In summary, empirical equations are suitable for simple, well-known blasting conditions, especially in scenarios where real-time performance is critical, but they have lower accuracy and limited applicability. ML predictions, on the other hand, excel in handling complex and variable conditions, offering higher accuracy and adaptability, but require large datasets and computational resources with poorer model interpretability.

The hybrid prediction model that integrates empirical equations with ML techniques has significant advantages. While empirical equations are simple in form, they capture key physical relationships in the field of blast-induced vibrations, such as the impact of factors like Q and R on vibration. On the other hand, ML techniques demonstrate outstanding ability in simulating the nonlinear interactions between complex variables. Therefore, combining empirical equations, such as the Sadowski equation, with advanced artificial intelligence techniques (such as ANN, SVM, or GEP) allows for the full utilization of the interpretability of empirical equations and the adaptability of ML models. By training hybrid models on multi-source datasets that include empirical coefficients and field measurement data, the model can better adapt to different geological and blasting conditions. Moreover, combining physics-based equations with ML outputs, such as using SHAP values to explain the predictions of ML models, can further clarify the specific contribution of individual factors to the vibration frequency. To advance the development of this field, it is recommended to establish large-scale, diverse databases to train and validate hybrid models in different blasting scenarios, continually improving their prediction performance and generalization ability.

6. Discussion

As a key characteristic parameter of blast-induced vibration, the vibration frequency is also an important basis for assessing the safety of blasting operations. This paper provides a systematic review and summary of relevant literature, focusing on the current research status, development trends, and key issues in areas such as blast-induced vibration safety assessment standards, frequency attenuation laws, and influencing factors, and dominant frequency prediction.

This paper first explains the importance of PPV and frequency as key indicators for assessing blast-induced vibration safety and provides a systematic summary of the safety standards for blast-induced vibration adopted by various countries. However, inconsistencies exist among the standards of different countries, and there is a general lack of evaluation on the impact of blast-induced vibrations on structures throughout their entire lifecycle and their effects on human psychology. The paper then outlines the characteristic frequencies in blast-induced vibration spectral analysis, including the zero-cross dominant frequency, Fourier dominant frequency, and Centroid frequency, and points out the current lack of a systematic analysis of the applicability of these three dominant frequencies. Finally, the paper summarizes the evolution of blast-induced vibration spectral analysis techniques from the Short-Time Fourier Transform (STFT) to the Wavelet Transform and then to the Hilbert–Huang Transform (HHT). These signal processing techniques progressively overcome the limitations of traditional Fourier Transform, enabling more adaptive, accurate, and comprehensive analysis of instantaneous energy and frequency characteristics for nonlinear, non-stationary signals.

The dominant frequency of blast-induced vibrations is influenced by multiple factors. When the charge amount is large, it leads to a lower dominant frequency because higher explosive energy releases vibrations at lower frequencies. As R increases, the dominant frequency typically decreases due to the attenuation of high-frequency energy during propagation. The initiation method also has a significant impact; sequentially delayed initiation typically increases the dominant frequency, thereby reducing the impact on surrounding structures. Smaller boreholes generate higher dominant frequencies while increasing the borehole diameter leads to a decrease in the dominant frequency. An increase in the number of free surfaces helps transfer vibrational energy to higher frequency bands, reducing the impact on buildings. Geological conditions, such as rock type and terrain, also affect wave propagation characteristics, with hard rock materials typically generating higher dominant frequencies. An increase in tectonic stress results in low-frequency energy dominating, especially under high-stress conditions. However, current research on the influencing factors of vibration frequency is relatively limited, and in-depth studies on key factors such as drilling parameters, free surfaces, burden, explosive structure, and delay time are scarce. Additionally, the specific influence mechanisms of parameters such as coupling coefficient, blasthole density coefficient, and explosive unit consumption on the dominant frequency need further exploration. Factors such as hole depth, hole spacing, vibration propagation path, and seismic source type also affect the dominant frequency. Deeper holes may change the stress distribution and wave propagation patterns, thus impacting the frequency; larger hole spacing typically leads to a lower frequency, while smaller hole spacing may increase the frequency. Moreover, the medium in the vibration propagation path (e.g., rock fractures, humidity, and temperature changes) and the shape of the seismic source (e.g., spherical or cylindrical) also affect the dominant frequency. Existing studies are mostly based on empirical and case analyses, lacking scientific evidence and extensive validation, leading to unreliable results. Therefore, in order to improve the safety and efficiency of blasting operations, there is an urgent need for more in-depth and comprehensive scientific research to clarify the impact of various parameters on blast-induced vibration dominant frequency, providing a scientific basis for optimizing blasting design.

In the fields of mining engineering, construction engineering, and environmental protection, the application of dominant frequency prediction models demonstrates significant value. In mining, by optimizing blasting design, such as adjusting charge weight, delay time, and drilling parameters, specific optimization plans can be proposed to avoid resonance with sensitive equipment and reduce the risk of damage. In construction engineering, frequency monitoring is implemented for different types of buildings to avoid resonance risks, and an analysis of the structural full-life-cycle vibration accumulation effect is introduced to comprehensively assess safety standards. In environmental protection, blasting designs with high-frequency attenuation structures are adopted to reduce low-frequency vibration impacts in ecologically sensitive areas, while frequency regulation is combined to lower noise pollution. ML models are employed to achieve multi-objective optimization, coordinating noise and vibration control, thereby comprehensively protecting vegetation and ecosystems. Based on the literature review, empirical equations are still widely used for predicting dominant frequency due to their simplicity. Savage’s equation only considers R for predicting dominant frequency, while other empirical formulas take into account a maximum of three parameters. However, there are numerous factors that influence the dominant frequency of blasting, which makes the accuracy of empirical equations lower compared to other methods. Additionally, due to the lack of uniformity in empirical equations, it is challenging to select one as the standard equation for predicting dominant frequency. Furthermore, existing research results are summarized, and a correlation assessment is conducted for the parameters that affect the prediction of dominant frequency. Parameters with high correlation are considered in the empirical formulas. Efforts should be made to provide a unified empirical equation that is suitable for most geological conditions and has relatively small prediction errors.

This study is consistent with earlier research in recognizing that the frequency of blast-induced vibrations is influenced by various factors, including the type of explosive, charge weight, borehole diameter, free surface conditions, burden, and geological conditions. However, this study further delves into the specific effects of these factors on vibration frequency, revealing the complex interactions between them. While earlier studies proposed prediction methods based on empirical formulas, this study highlights the advantages of ML in predicting blast-induced vibration frequencies. By comparing traditional methods with the latest advancements in ML applications, the study reveals the significant potential of ML in improving prediction accuracy. Empirical equations use a large amount of data for regression and variance analysis, with different empirical coefficients applied to different geological conditions based on rock types. Typically, these empirical coefficients fall within a reasonable range according to the fitting results, but the issue of low prediction accuracy remains. Therefore, incorporating additional geological features, such as porosity, rock hardness, fault structures, and other factors, which can significantly influence the propagation of blast vibrations and dominant frequency, is necessary. By integrating these variables, the applicability of the empirical equations can be improved, reducing errors caused by varying geological conditions. A more detailed classification of blast data under different regions or geological conditions can lead to the establishment of localized empirical formulas. Expanding the dataset with diverse data and combining different types of models (such as empirical equations and ML) will further enhance the robustness and accuracy of predictions. Additionally, data augmentation using existing vibration data (such as adding synthetic or simulated data) can help the model better adapt to different geological and blasting conditions.

The correlation coefficients between the predicted values of the ANN and ANFIS models and the measured data are 0.95 and 0.9988, respectively, indicating that these two models exhibit high prediction accuracy. Additionally, the SVM model, when predicting the dominant frequency, shows that the relative error is kept within 10% in most cases, demonstrating its efficiency and accuracy in prediction tasks. These results strongly validate the robust predictive capability of the proposed models, highlighting the significant advantages of ML methods in this study and providing solid support for applications in related fields. Although ML methods can significantly improve prediction accuracy, issues such as insufficient sample size and poor generalization ability may lead to reduced prediction accuracy. For example, in reference [116], the average relative errors for BP neural network prediction and empirical formula prediction were 13.59% and 11.16%, respectively, showing that ML exhibits lower prediction accuracy compared to empirical formula methods. When selecting a prediction method, one should balance data availability, model complexity, and the accuracy requirements in practical applications.

Engineers should comprehensively evaluate the dominant frequency of blast-induced vibrations along with PPV to more accurately predict the impact of blasting on building structures. Special attention should be given to resonance risks. For buildings with natural frequencies close to the dominant frequency of blast-induced vibrations, measures to reduce energy transfer, such as adjusting charge weight or delay time, should be taken. Based on different geological conditions and building types, reasonable thresholds for dominant frequency and PPV should be established to ensure structural safety. For different rock types, optimize borehole spacing, charge weight, and delay time to reduce unnecessary vibration propagation and improve blasting efficiency. By combining time–frequency analysis methods such as wavelet transform, real-time monitoring of vibration signals can be conducted, and blasting plans can be adjusted based on data to reduce adverse impacts on the surrounding environment. Using optimal delay time design, the explosive sequence can be controlled to lower overall vibration intensity. Sensor networks should be deployed at construction sites to monitor the frequency and intensity of blast-induced vibrations in real time. By integrating ML algorithms or signal processing techniques, potential high-risk vibration zones can be predicted, and subsequent blasting plans can be optimized in advance. Additionally, safety training for engineers and operators should be strengthened, particularly regarding the potential risks of vibrations to structures, personnel, and the environment.

7. Outlook

This paper provides an outlook based on a literature review, addressing the existing shortcomings in current blast vibration safety standards, the factors influencing dominant frequency, and the predictive methods for blast-induced vibrations:

• There are significant differences in blasting vibration safety standards across different countries, particularly regarding the setting of frequency-velocity thresholds. For example, the standards in the United States, Switzerland, and Germany show inconsistencies in building classification and frequency division, while the standards in Italy and China may underestimate the risks associated with high-frequency vibrations in the high-frequency band. To promote the safety and sustainability of global blasting operations, there is an urgent need to establish a unified evaluation framework. Future research may require international organizations to coordinate data from multiple countries and develop a hierarchical evaluation system based on structural types, geological conditions, and service life. To more effectively control the impact of blast-induced vibrations on surrounding structures, it is first necessary to conduct a natural frequency survey of protected targets (such as buildings and equipment) and establish a relevant database. Before blasting, the natural frequencies of nearby structures should be assessed, and parameters such as charge weight and delay time should be dynamically adjusted to ensure that the dominant frequency is far from the sensitive frequency range of the structures. Additionally, by deploying a sensor network to collect real-time data on PPV and dominant frequency from blasting vibrations and combining ML models (such as PSO-LSSVM and random forests), vibration propagation can be predicted. On this basis, an adaptive control system can be developed, capable of automatically alarming when the dominant frequency approaches the structure’s natural frequency, allowing adjustments to blasting parameters to ensure safety. Furthermore, it is essential to define the applicable scenarios for different dominant frequency types. For engineering analysis, zero-cross dominant frequency should be prioritized, while dynamic response evaluation should use the response spectrum dominant frequency. For complex signals, wavelet transform or HHT analysis must be performed to identify multi-peak frequency components.
• The dominant frequency of blast-induced vibration is a key indicator for assessing the impact of blasting operations on the environment and structures. When the dominant frequency approaches the natural frequency of a structure, it may pose a threat to the safety of buildings and personnel. Therefore, strict control of vibration frequency is necessary. Currently, there are no clear standards for the applicability of Zero-cross dominant frequency, Fourier dominant frequency, and Centroid frequency, and further research is required.
• In the future, the analysis of blast-induced vibration signals should integrate various time–frequency analysis methods (such as Fourier Transform, Wavelet Transform, and HHT) to construct a comprehensive analysis framework. With the aid of ML and intelligent algorithms, automatic feature recognition should be achieved, and real-time monitoring and early warning of vibration signals should be realized through the integration of high-precision monitoring equipment.
• An increase in the number of delay segments causes the vibration signal energy to concentrate in the mid-high frequency range, resulting in an increase in the dominant frequency. An increase in R and burden leads to a decrease in the dominant frequency. Additionally, increases in hole diameter, the number of free surfaces, maximum charge quantity, and elevation raise the dominant frequency. Furthermore, the hardness, density, and elastic modulus of the rock, as well as the damping effect of the medium, significantly affect the vibration propagation characteristics. However, research on factors such as borehole parameters, free face, charge coefficient, and delay time remains limited and requires further exploration. An in-depth study of these factors is essential to achieve precise blasting control, avoid excessive material or energy consumption, and ultimately enhance the sustainability of the blasting industry. Future research could systematically study the impact of geological conditions, blasting parameters, and other factors on vibrations by designing different types of blasting experiments. For example, variables such as Q, R, and delay time could be varied in the experiment while controlling geological conditions (e.g., rock type, porosity, stratification), and the effects of these factors on vibration frequency, amplitude, and attenuation patterns could be observed. Reliable data collected from multiple experiments would provide a solid foundation for subsequent model development. In terms of field measurements, it is recommended to use high-precision vibration sensors for multi-point synchronized measurements, particularly in different locations near and far from the blast source, to ensure comprehensive capture of frequency variations during the vibration propagation process. At the same time, integrating real-time monitoring systems to dynamically track the time–frequency characteristics of vibrations would help acquire richer vibration data. Advanced technologies such as 3D seismic exploration and fiber-optic sensors could be used for data collection, providing more comprehensive geological information and blasting data. In terms of data modeling methods, combining ML, deep learning, and other advanced technologies can improve the predictive accuracy for complex blast vibration characteristics. For example, ML methods could be used to train and validate predictive models for blast vibrations, integrating traditional physical models with modern statistical methods to offer more comprehensive support for vibration prediction models, thereby enhancing the reliability and accuracy of prediction results.
• Existing dominant frequency equations are primarily based on the Sadove formula, which only considers a few factors, such as R and charge quantity, limiting their prediction accuracy and applicability. In contrast, ML algorithms have significantly improved the accuracy of dominant frequency prediction, although issues related to algorithm complexity and limitations still need to be addressed. In the future, a large-scale database should be established, widely applicable for predicting open-pit mining blasts, to prevent the fundamental frequency from approaching the natural frequency of structures, reducing the potential threats of blasting to the environment and buildings, and ensuring the sustainable development of blasting engineering.
• To enhance the accuracy and safety of the blasting design, multiple influencing factors (such as charge weight, charge structure, initiation sequence, geological conditions, and R) are comprehensively considered and incorporated into the dominant frequency prediction model. Based on the prediction results, engineers can adjust the charge weight and charge structure to achieve the ideal vibration frequency range, thereby reducing the risk of damage to surrounding structures. Optimizing the initiation sequence and delay time helps to reduce the superimposition effect of vibration waves, further decreasing the impact on the surrounding environment. Geological conditions play an important role in the propagation and attenuation of vibration frequencies. By considering the influence of geological conditions, engineers can more accurately assess the variation of vibration frequencies under different geological conditions and adjust the blasting design to suit the specific geological environment.

In summary, the future research directions are summarized in

Table 10. Future research.

Research Field	Future Research Directions
Safety Standards	By considering multiple factors, establish a unified evaluation framework based on structural types and geological conditions.
Dominant Frequency Type	Through theoretical and experimental studies, establish application criteria for different dominant frequency types and clarify their applicability in engineering scenarios.
Signal Analysis Techniques	Construct a time–frequency analysis framework based on multi-method integration; combine ML to achieve intelligent signal feature recognition and real-time monitoring and early warning.
Influencing Factors	Conduct in-depth research on the influence mechanisms of complex factors (such as geological conditions, charge structure, and uncoupling coefficient) on the dominant frequency; explore the patterns of difficult-to-quantify parameters.
Frequency Prediction	Consider the impact of geological conditions and establish a large-scale database, widely applicable for predicting open-pit mining blasting; comprehensively consider multiple factors and incorporate them into dominant frequency prediction to improve the accuracy and safety of blasting design; adjust the blasting design based on predictions, optimize blasting parameters, and ensure operational safety.

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