A Research on the Delay Time of Vibration Reduction in an Open Pit Mine Based on the Dominant Principal Frequency

Abstract

Vibrations generated by blasting operations in open-pit mines can have adverse effects on nearby buildings. To mitigate these vibrations by selecting an appropriate delay time, blasting vibration monitoring experiments were conducted in an open-pit mine. The characteristics of peak particle velocity (PPV) and principal frequencies of seismic waves from single-hole and group-hole blasting were analyzed, and an equation for the decay of PPV during the propagation of blasting seismic waves was established. To address the uncertainty in the principal frequency of seismic waves from single-hole blasting, the dominant principal frequency of single-hole blasting seismic waves was defined. A method for determining the optimal vibration-reduction delay time based on the dominant principal frequency of single-hole blasting seismic waves was proposed. Using the bootstrap method, the dominant principal frequency of single-hole blasting seismic waves was estimated to be 20 Hz. Combining this with the principle of staggered-phase superposition, the optimal vibration-reduction delay time between boreholes was determined to be 25 ms. The vibration-reduction effectiveness of the optimal delay time was verified through field experiments. The results show that, compared with the original inter-hole delay time of 42 ms, a delay time of 25 ms achieves good vibration reduction, with average reduction rates of 28.3%, 19.4%, and 20.6% in the horizontal radial, horizontal tangential, and vertical directions, respectively. These research findings can serve as a reference for optimizing delay times and controlling blasting vibrations in deep-hole open-pit blasting.

1. Introduction

Blasting is an efficient and economical technique for rock excavation in open-pit mining. However, only part of the explosive energy contributes to rock fragmentation, while much of it is dissipated as unwanted environmental effects, particularly ground vibration [1,2]. Seismic waves generated by blasting propagate along the ground surface and can induce vibration responses in nearby buildings and structures [3,4], such as concrete buildings [5], buried pipeline [6] and historical building [7]. Such blast-induced vibration may cause structural damage or even failure, raising concerns among residents living near mining areas [8].

Traditional vibration-mitigation measures have been widely adopted to reduce blast-induced vibration and protect nearby structures in mining areas. Common methods include vibration-damping trenches and isolation pits, which interrupt or weaken the propagation of vibration waves, with geosynthetic foam often used as a barrier-filling material [9,10,11]. Field tests and numerical simulations have confirmed the effectiveness of these measures in isolating continuous vibration and resonance [12]. Nevertheless, their use is limited by site conditions, construction space, and the need for additional excavation and installation work, which can substantially increase project costs. In this regard, vibration reduction through the optimisation of blasting parameters provides a more economical and practicable alternative.

Garai et al. [13] showed that peak particle velocity (PPV) is closely related to borehole geometry. In field drilling and blasting operations, the detonation direction and initiation sequence can change the superposition angle of vibration waves, causing PPV to vary considerably. Roy et al. [14] found that, at short distances, blast-induced vibration depends mainly on the total charge quantity, while its relationship with the maximum charge per delay is relatively weak. Hong et al. [15] reported that the coupling medium strongly affects the ground vibration response. Among the tested cases, clay blasting under air-decoupled conditions generated the lowest ground vibration and caused the least disturbance to the surrounding rock mass, buildings, and structures. Silveira et al. [16] reduced blast-induced vibration intensity by optimising the borehole diameter. Rezaeineshat et al. [17] reported that PPV was reduced by up to 59% when the hole spacing was set to 4.4 m and the maximum charge per delay to 627 kg. Sun [18] found that the attenuation rate of the principal frequency increased as the borehole diameter or decoupling coefficient decreased, with borehole diameter having the stronger effect. Taken together, these studies show that borehole diameter, coupling medium, charge configuration, and charge weight are key parameters for controlling blast-induced vibration.

The vibration-mitigation effect of blasting parameters is mainly achieved by reducing or adjusting the energy released at the blast source. Although this approach can reduce vibration intensity, it may also affect rock fragmentation efficiency. In contrast, delay time does not directly change the energy released from a single-hole blasting. Rather, it controls the initiation interval between multi-stage or group-hole blasts, thereby altering the temporal superposition of vibration waves and the distribution of energy in the frequency domain. Through destructive interference and local peak attenuation, an appropriate delay time can reduce blast-induced vibration without substantially changing the energy scale of individual blastholes [19]. For this reason, delay-time optimisation provides a useful means of balancing blasting performance and vibration control.

The wider use of digital electronic detonators in open-pit blasting has made it possible to set delay time more precisely and to apply vibration reduction based on staggered-phase superposition [20,21]. Iwano et al. [22] conducted vibration-reduction tests for tunnel blasting using advanced electronic detonators. Their research showed that the optimal delay time could be estimated based on single-hole blasting waveforms, and this was verified at another tunnel site. Wang et al. [23] determined delay time from the distribution of instantaneous energy peaks obtained by the Hilbert-Huang transform (HHT). Tao et al. [24] used linear superposition to show that the calculated delay time was 10–15 ms, which was longer than that obtained from field tests. Lin et al. [25] predicted seismic waveforms for single-blast operations based on linear superposition and identified the optimal delay time by comparing PPV under different delay times. Wang et al. [26] further developed a seismic-wave superposition method for delayed blasting that accounts for changes in free surfaces, allowing more accurate prediction and control of vibration in bench blasting. Agrawal et al. [27] analysed single-hole blasting seismic waveforms and determined the optimal inter-hole and inter-row delay times as 25 ms and 67 ms, respectively. Yang et al. [28] suggested that group-hole seed waves can be used to control blast-induced vibration under different blasting conditions. Wang et al. [29] adjusted local delay times to separate vibration waves before their convergence, achieving a vibration reduction rate of 14.05% while maintaining blasting effectiveness.

The core of the above method lies in selecting representative seismic waves from single-hole blasting. The vibration-reduction delay time is obtained by analyzing single-hole blast seismic waves, or by performing a superposition analysis of these waves. However, during blasting operations, the waveforms of single-hole blast seismic waves vary due to some factors. Selecting different single-hole blast seismic waves yields different vibration-reduction delay times. This adversely affects the accuracy of the vibration-reduction delay times determined by such methods.

To address the uncertainty in optimizing the vibration-reduction delay time caused by variations in single-hole blast seismic waveforms, this paper employs maximum likelihood estimation, Bayesian analysis, and Bootstrap method to analyze the characteristics of the principal frequency distribution of single-hole blast seismic waves. The Bootstrap method was employed to estimate the confidence intervals for the principal frequencies of single-hole blasting seismic waves. The frequency within the 95% confidence interval is defined as the dominant principal frequency. The half-cycle corresponding to the dominant principal frequency was used as the vibration-reduction delay time, and the effectiveness of this method in reducing blast vibrations was verified using the practical applications in blasting engineering. This method avoids the difficulty of selecting a single-hole blasting seismic waveform and achieves good vibration reduction effects from blasting.

2. Methods

2.1. Superposition of Vibration Waves

Blasting seismic waves can be approximated as simple harmonic waves. According to the principle of superposition of waves, when two groups of blasting seismic waves have a certain phase difference, their crests and troughs may partly or completely cancel each other, thereby reducing blast-induced vibration. It reduces vibration by optimising the inter-hole delay time, which adjusts the phase difference between seismic waves from different holes when they arrive at the monitoring point.

When the geological conditions between the blasting area and the monitoring point are broadly consistent, and the charge quantity in each borehole is the same, the vibration responses generated by different holes at the monitoring point are highly similar. The resulting blasting vibration effect can be expressed as follows [19]:

$$A\left(t\right)={\displaystyle \sum _{i=1}^n}{A}_i\left(t\right)$$

(1)

where $A\left(t\right)$ is the vibration velocity at the measurement point; $A_i\left(t\right)$ is the vibration velocity generated by the single-hole blasting; $t$ is the time; $i=\mathrm{1,2},3,\dots n$.

When blasting seismic waves are treated as harmonic waves, they can be expressed as follows [30]:

$$A_i\left(t\right)=\sin \left(\omega t-{\phi }_i\right)$$

(2)

where $\sin \left(\omega t-{\phi }_i\right)$ represents the harmonic wave; $\omega$ is the angular frequency; ${\phi }_i$ is the phase of the wave.

For the superposition of vibration waves from two holes, the expression is given by:

$$A_i=\sin \left(\omega t-{\phi }_1\right)+\sin \left(\omega t-{\phi }_2\right)=2sin\left(\omega t-{\displaystyle \frac{{\phi }_1+{\phi }_2}{2}}\right)\cos \left({\displaystyle \frac{{\phi }_1-{\phi }_2}{2}}\right)$$

(3)

where ${\phi }_1$ is the phase of the seismic wave generated by the first borehole; ${\phi }_2$ is the phase of the seismic wave generated by the second borehole.

For the resultant vibration velocity to be lower than that of a single-hole blast, the following condition must be satisfied:

$$-{\displaystyle \frac{1}{2}}<\cos \left({\displaystyle \frac{{\phi }_2-{\phi }_1}{2}}\right)<{\displaystyle \frac{1}{2}}$$

(4)

$${\displaystyle \frac{\pi }{3\omega }}<{\displaystyle \frac{{\phi }_2-{\phi }_1}{2}}<{\displaystyle \frac{2\pi }{3\omega }}$$

(5)

Taking the principal oscillation period of the seismic wave from blasting to be $T=1/f$, the separation between the two wave groups follows Equation (6).

$$kT+{\displaystyle \frac{T}{3}}<∆t<kT+{\displaystyle \frac{2T}{3}}$$

(6)

where $k$ is a non-negative integer; $∆t$ is the inter-hole delay time of the seismic wave; $T$ is the main oscillation period, $f$ is the principal frequency.

Under this condition, the seismic waves can cancel each other to varying degrees. When $∆t=kT+{\displaystyle \frac{T}{2}}$, the vibration-reduction effect reaches its maximum.

Figure 1 illustrates the superposition of seismic waves.

2.2. Bootstrap Method

As a typical nonparametric statistical method, the bootstrap method offers good stability when dealing with small sample sizes. The calculation steps are as follows [31]:

(1) Construct an empirical distribution function $F_n\left(x\right)$ based on a random sample $X = \{x_1,{ x}_2, ..., x_n\}$. The equation is shown below.

$${ F}_n\left(x\right)=\left\{\begin{array}{cc}0& x<x_{\left(1\right)}\\ {\displaystyle \frac{i}{n}}& x_{\left(1\right)}<x<x_{\left(n\right)}\\ 1& x>x_{\left(n\right)}\end{array}\right.$$

(7)

(2) Perform n rounds of resampling with replacement on the original data. This yields a Bootstrap sample $X_1^{*} = \left\{x_{11}, x_{21}, ..., x_{n1}\right\}$.

(3) Repeat the resampling process from Step (2) independently m times (m ≥ 1000). This yields the matrix shown in Equation (8).

$$X^{*}=\left(X_1^{*},X_2^{*},\dots ,X_m^{*},\right)=\left[\begin{array}{cccc}{X}_{11}^{*},& X_{12}^{*},& \dots ,& X_{1m}^{*}\\ X_{21}^{*},& X_{22}^{*},& \dots ,& X_{2m}^{*}\\ ⋮& ⋮& ⋮& ⋮\\ X_{n1}^{*},& X_{n2}^{*},& \dots ,& X_{nm}^{*}\end{array}\right]$$

(8)

(4) Calculate the mean ${\stackrel{-}{X}}_j^{*}$ and standard deviation $S_j^{*}$ of sample $X_j\left(j=1,2,\dots ,m\right)$. These values give rise to the Bootstrap sampling distributions of the mean $\widehat{\mu }$ and the standard deviation $\widehat{\sigma }$.

(5) Using the Bootstrap quantile method at a significance level α = 0.05, we estimate the mean and variance of the unknown parameter θ via interval estimation, thereby obtaining two-sided 95% confidence intervals.

2.3. Determining the Vibration-Reduction Delay Time Using the Dominant Principal Frequency

The dominant principal frequency is defined as the centre point of the 95% confidence interval for the mean principal frequency of seismic waves from a single-hole blast, as determined through Bootstrap analysis. According to the theory of staggered-phase blasting vibration reduction, the optimal delay time falls within range $kT+{\displaystyle \frac{T}{3}}<∆t<kT+{\displaystyle \frac{2T}{3}}$, and the best results are achieved when $∆t=kT+{\displaystyle \frac{T}{2}}$. Therefore, the half-cycle corresponding to the dominant principal frequency is determined to be the optimal vibration reduction delay time. The methodology for determining the optimal delay time based on the dominant principal frequency is shown in Figure 2.

3. Characteristics of Seismic Waves from Single-Hole and Group-Hole Blasting

3.1. Monitoring of Blasting Vibration

The experiment was conducted at an open-pit mine that uses the deep-hole blasting method for overburden removal and ore extraction. Charging trucks are used to load ammonium nitrate fuel oil explosives and emulsion explosives into the holes, and digital electronic detonators are used for hole-by-hole initiation. The holes are arranged in a diamond pattern, with an inter-hole spacing of 4 m and an inter-row spacing of 3 m. The stemming length is 4 m, and the maximum charge per borehole is 300 kg, using a continuous charging structure.

The experimental blasting area is located on the western slope of the mine. Monitoring points were arranged along a straight line at distances of 25–300 m from the blasting area. Before production blasting, vibration meters were installed at these monitoring points to record blasting seismic waves. The triaxial sensors were fixed to the ground with steel rods, with their X-axes oriented towards the blasting area. The layout of the blast-induced vibration monitoring points is shown in Figure 3.

A schematic diagram of the vibration monitoring setup is shown in Figure 4. The monitoring system consists of a TC-4850N blast vibration analyser, a triaxial vibration velocity sensor, a computer, and a data analysis platform. The blast vibration analyser has a frequency range of 5–500 Hz, a sampling rate of 100–100,000 Sps, a resolution of 0.01 cm/s, and a maximum measurement range of 35 cm/s. The triaxial vibration velocity sensor can record blasting seismic waves in three directions simultaneously, with a frequency response range of 1–500 Hz and a sensitivity of 26–28 V/(m/s).

3.2. PPV Characteristics in Single-Hole and Group-Hole Blasting

Determining the optimal inter-hole delay time requires an analysis of the seismic waves generated by single-hole blasting. After several field trials, the final borehole was detonated 300 ms after the last borehole in the group, so that a single-hole blasting seismic wave could be obtained. The measured seismic waves from group-hole and single-hole blasting are shown in Figure 5. As shown in Figure 5, the waveform of the final borehole is clearly separated from that of the group holes, with an interval of approximately 300 ms. This agrees with the blasting design. Compared with the single-hole waveform, the group-hole waveform is more complex and has a higher peak particle velocity. For both group-hole and single-hole blasting, the peak particle velocity is highest in the horizontal tangential direction, followed by the horizontal radial direction, and lowest in the vertical direction.

Figure 6 shows the distribution of PPV for single-hole and group-hole blasting at different distances from the blast centre. The PPV in the horizontal directions is higher than that in the vertical direction for both blasting types. In all three directions, the PPV exhibits a decreasing trend with increasing distance.

A regression analysis of PPV was performed using the Sadovsky formula to determine the attenuation characteristics of blasting seismic waves. The Sadovsky formula is expressed as follows:

$$V=K{\left({\displaystyle \frac{\sqrt[3]{Q}}{R}}\right)}^{\alpha }$$

(9)

where $V$ is the PPV; $Q$ is the maximum charge per delay; $R$ is the distance from the monitoring point to the blast source; $K$ is the site coefficient; $\alpha$ is the vibration attenuation exponent.

Taking the logarithm of both sides of Equation (9) gives Equation (10):

$$\ln V=\ln K+\alpha \ln \left({\displaystyle \frac{\sqrt[3]{Q}}{R}}\right)$$

(10)

When $y=\ln V$, $a=\alpha$, $x=\ln \left({\displaystyle \frac{\sqrt[3]{Q}}{R}}\right)$, $b=\mathit{ln}K$, Equation (10) can be rewritten as:

$$y=kx+b$$

(11)

Figure 7 shows the fitting results of PPV in different directions using the Sadovsky formula. The coefficient of determination (R²) is greater than 0.8 in all three directions, indicating that the formula fits the measured data well. The highest R² is obtained in the vertical direction, with a value of 0.90, while the lowest value is 0.81 in the horizontal tangential direction.

The attenuation relationships for blasting seismic waves in the horizontal radial, horizontal tangential, and vertical directions are given in Equations (12)–(14). These equations show that the PPV in all three directions decreases as the propagation distance increases. The vertical PPV attenuates faster than the horizontal radial and horizontal tangential PPVs, while the attenuation rates in the two horizontal directions are relatively similar.

$$V_X=144.03{\left({\displaystyle \frac{\sqrt[3]{Q}}{R}}\right)}^{1.38}$$

(12)

$$V_Y=95.58{\left({\displaystyle \frac{\sqrt[3]{Q}}{R}}\right)}^{1.20}$$

(13)

$$V_Z=137{\left({\displaystyle \frac{\sqrt[3]{Q}}{R}}\right)}^{1.69}$$

(14)

3.3. Principal Frequency Characteristics in Single-Hole and Group-Hole Blasting

Table 1. Statistical indicators of principal frequencies in single-hole and group-hole blasting seismic waves.

	Direction	Minimum/Hz	Maximum/Hz	Mean/Hz	Standard Deviation/Hz
Group-hole	Horizontal radial	13.072	32.00	22.492	4.435
	Horizontal tangential	13.158	40.404	23.182	5.973
	Vertical	14.599	38.095	23.160	5.814
Single-hole	Horizontal radial	10.526	34.483	21.957	5.061
	Horizontal tangential	10.84	32.258	22.88	4.884
	Vertical	10.336	34.783	21.445	5.602

gives the statistical indicators of the principal frequencies of seismic waves from single-hole and group-hole blasting. The indicators differ only slightly between the two blasting types. This may be because the superposition of wave peaks in group-hole blasting mainly amplifies the vibration amplitude, while the principal frequency of the superimposed seismic waves remains close to that of single-hole blasting seismic waves.

Figure 8 shows the distribution of the principal frequencies of seismic waves from single-hole and group-hole blasting. In different directions, the principal frequencies for both blasting types approximately follow a normal distribution. Most values are concentrated between 10 and 40 Hz, while frequencies below 10 Hz and above 40 Hz account for only a small proportion. The dominant principal frequency range is 20–25 Hz.

4. Determination of the Optimal Inter-Hole Delay Time

14 groups of measured principal frequencies of single-hole blast waves were selected as the sample for analyzing the optimal delay time between holes. The samples were all derived from field measurements taken under similar engineering geological conditions and blasting parameters. They can represent the primary characteristics of single-hole blast seismic waves under the conditions of this project. The principal frequencies for the horizontal radial (X₁), horizontal tangential (X₂), and vertical (X₃) directions are given as follows:

• X₁ = (10.363, 15.873, 17.857, 20.513, 19.9, 22.346, 18.957, 20.202, 30.534, 27.027, 24.39, 21.505, 18.779, 25.478);
• X₂ = (21.739, 16.26, 24.691, 21.277, 24.54, 20.833, 19.512, 27.778, 25.157, 22.599, 10.23, 23.256, 18.692, 20.833);
• X₃ = (14.184, 17.778, 25.157, 10.336, 26.49, 19.472, 10.61, 18.265, 20.408, 22.222, 23.121, 13.699, 16.064, 22.472).
• unit: Hz

The Bootstrap method is a typical non-parametric statistical method that is robust for small-sample analysis and can quantify uncertainty not captured by point estimates. In this research, the principal frequency samples in the horizontal radial direction from single-hole blasting were used as an example, and the other samples were analysed in the same way. The dominant principal frequency of single-hole blasting seismic waves was then determined using this method.

Figure 9 shows the variation in the mean of the Bootstrap sample means for X₁ with the number of Bootstrap samples. When the sample size exceeds 1000, the mean becomes stable. Figure 10 shows the Q–Q plot of the Bootstrap sample means for X₁. The Bootstrap sample means follow a normal distribution, N(20.977,1.332), which is close to the mean of the original X₁ sample, 20.98 Hz. This indicates that the Bootstrap samples retain the main characteristics of the original sample.

The confidence interval for sample X₁ was estimated using the percentile method. The 95% confidence interval for the population mean of X₁ was [18.85, 23.12] Hz. The same procedure was applied to samples X₂ and X₃ to estimate the confidence intervals of their principal frequencies.

Table 2. Confidence intervals for the mean principal frequencies of single-hole blasting seismic waves in different samples.

Sample	Sample Mean/Hz	95% Confidence Interval/Hz
X1 (n = 14)	20.98	[18.85, 23.12]
X2 (n = 14)	21.23	[19.23, 23.01]
X3 (n = 14)	18.60	[16.29, 20.78]

lists the confidence intervals for the mean principal frequencies of single-hole blasting seismic waves in different directions.

To verify the accuracy of the Bootstrap method in estimating the confidence intervals for the mean principal frequencies of single-hole blasting seismic waves, the results were compared with those obtained using maximum likelihood estimation and Bayesian analysis. Maximum likelihood estimation is one of the most commonly used and efficient methods for parameter estimation in statistics. It is defined as follows [32]:

Let the population distribution be $P\left(X;\theta \right)$, where $\theta$ is the parameter to be estimated. A set of independent and identically distributed sample functions is given by Equation (15):

$$L\left(\theta ;X_1,\dots ,X_n\right)={\displaystyle \prod _{i=1}^n}P\left(X_i;\theta \right)$$

(15)

The maximum likelihood estimation method aims to find a parameter estimate ${\widehat{\theta }}_{MLE}$ such that the likelihood function attains its maximum value at that point.

$${\widehat{\theta }}_{MLE}=arg \underset{\theta }{\max } L\left(\theta ;X_1,\dots ,X_n\right)$$

(16)

The confidence intervals estimated by maximum likelihood estimation and the corresponding data histograms are shown in

Table 3. Confidence interval estimates of the mean principal frequency obtained by maximum likelihood estimation.

Sample	Sample Mean/Hz	95% Confidence Interval/Hz
X1 (n = 14)	20.98	[18.21, 23.75]
X2 (n = 14)	21.24	[18.84, 23.65]
X3 (n = 14)	18.59	[15.74, 21.44]

and Figure 11, respectively.

As shown in Figure 11, the dominant principal frequency in the horizontal radial direction is 20.98 Hz, and the principal frequencies are mainly concentrated between 17 and 23 Hz. In the horizontal tangential direction, the dominant principal frequency is 21.24 Hz, with most values distributed between 17 and 25 Hz. In the vertical direction, the dominant principal frequency is 18.59 Hz, and the principal frequencies are mainly distributed between 16 and 24 Hz. The peak of the fitted normal distribution curve does not fully coincide with the peak of the histogram. This is because the peak position of the fitted normal curve is determined by the sample mean and standard deviation, whereas the histogram peak is more strongly affected by local fluctuations in sample frequency. This suggests that the estimates obtained by maximum likelihood estimation are relatively sensitive to sample fluctuations.

Finally, Bayesian analysis was used to further validate the main frequency estimates. The core of this approach is to combine prior knowledge with newly acquired data to derive a posterior distribution, thereby updating and optimizing the probability estimates.

The expression is shown in Equation (17) [33]:

$$P\left(\theta |f\right)={\displaystyle \frac{{\rm P}\left(f|\theta \right)P\left(\theta \right)}{P\left(f\right)}}$$

(17)

where $P\left(\theta \right)$ represents the prior distribution of the model parameters; $P\left(f|\theta \right)$ represents the sample likelihood function; $P\left(\theta |f\right)$ represents the posterior distribution after the parameters have been updated.

To reduce the influence of errors in the prior distribution on the estimation results, the mean and standard deviation of the principal frequencies obtained from Bootstrap resampling were used as the prior distribution for Bayesian analysis. The confidence interval estimates of the mean principal frequencies obtained by Bayesian analysis are shown in Figure 12. The corresponding confidence intervals for different samples are listed in

Table 4. Confidence interval estimates of the mean principal frequencies obtained by Bayesian analysis.

Sample	Sample Mean/Hz	95% Confidence Interval/Hz
X1 (n = 14)	20.98	[18.21, 23.75]
X2 (n = 14)	21.24	[18.84, 23.65]
X3 (n = 14)	18.59	[15.74, 21.44]

.

As shown in Figure 12, the principal frequencies approximately follow a normal distribution. Using the normal distribution as the prior distribution, the mean principal frequencies in the three directions are 20.98 Hz, 21.24 Hz, and 18.59 Hz, respectively.

Table 5. Confidence interval estimates of the mean principal frequencies obtained by different methods.

Sample	Bootstrap/Hz	Maximum Likelihood Estimation/Hz	Bayesian Analysis/Hz
X1 (n = 14)	[18.85, 23.12]	[18.21, 23.75]	[18.56, 23.40]
X2 (n = 14)	[19.23, 23.01]	[18.84, 23.65]	[19.14, 23.35]
X3 (n = 14)	[16.29, 20.78]	[15.74, 21.44]	[16.10, 21.09]

shows the results of confidence interval estimates for the mean principal frequency using different methods. A comprehensive comparison was conducted by combining the results of the Bootstrap method, maximum likelihood estimation, and Bayesian analysis. It was found that, at the 95% confidence level, the confidence intervals for the mean principal frequency obtained by the three methods were relatively close. The confidence interval for the principal frequency estimated using the Bootstrap method was the shortest, indicating that, under the conditions of the existing sample, the Bootstrap method produces estimates of the mean principal frequency with lower variability and greater stability. This suggests that the Bootstrap method can provide statistical support for determining the dominant principal frequency.

At a 95% confidence level, using the bootstrap method, the intersection of the confidence intervals for the mean principal frequencies of seismic waves from single-hole blasting in the horizontal radial (X₁), horizontal tangential (X₂), and vertical (X₃) directions was determined to be 19.23–20.78 Hz. The dominant principal frequencies are primarily distributed within the interval [19.23, 20.78]. Since 20 Hz is located near the center of this frequency interval, it adequately represents the dominant principal frequency component of the vibration signal from this blast. Therefore, this paper adopts 20 Hz as the dominant principal frequency for this single-hole blast in a deep open-pit mine.

According to the staggered-phase superposition theory of blasting seismic waves, when Δt = T/2, the time corresponding to a half-phase difference for each harmonic component within the principal frequency range is Δt = 1/(2f). Under this condition, the superimposed vibration velocity of the blasting seismic waves approaches zero, and the vibration-reduction effect is maximised. Substituting f = 20 Hz into Δt = 1/(2f) gives an optimal inter-hole delay time of 25 ms. Therefore, the closer the inter-hole delay time is to 25 ms, the better the vibration-reduction effect.

5. Engineering Application

5.1. Experimental Scheme

To verify the vibration-reduction effect of the 25 ms inter-hole delay time, an open-pit deep-hole blasting test was carried out on the eastern slope of the mine. In the test scheme, the blasting area contained 60 holes with a borehole diameter of 165 mm. The maximum charge per blasthole is 300 kg, the inter-hole spacing was 5 m, the inter-row spacing was 4 m, and the subdrilling depth was 2 m. The holes were arranged in a diamond pattern, and a continuous charging structure was adopted. Digital electronic detonators were used to control the initiation precisely. The inter-hole delay time in this scheme was 42 ms. A comparative test was then conducted at a nearby site with similar topographical and geological conditions. In the comparative test, the inter-hole delay time was set to 25 ms, while the other blasting parameters were kept the same as those in the test scheme. These two blasting areas are adjacent to each other and consist primarily of magnetite. The geometry of the blast surfaces and the free-surface conditions for the two blasts are similar. However, there are still some differences due to factors such as rock mass discontinuities. It should be noted that although the experimental conditions and blasting parameters are similar, the seismic waves generated by single-hole blasting are not entirely identical. Theoretically, these seismic waves can all produce vibration reduction effects through staggered-phase linear superposition. However, different seismic waves yield varying vibration reduction outcomes. Therefore, the linear superposition method is not recommended for validation.

The vibration-reduction effect under each delay time was evaluated using the vibration-reduction rate, which is defined as follows:

$$\delta ={\displaystyle \frac{V_1-V_2}{V_1}}\times 100\%$$

(18)

where $\delta$ is the vibration-reduction rate; $V_1$ is the PPV of the original group-hole blasting; $V_2$ is the PPV of group-hole blasting under a given inter-hole delay time.

According to the topographical and geological conditions of the mine site, three monitoring points were arranged near village houses on the ground surface close to the eastern slope of the mine. The TC–4850N vibration meter was used to record blasting seismic waves at different monitoring points. The layout of the field monitoring points is shown in Figure 13.

5.2. Analysis of the Vibration-Reduction Effect

The blast-induced vibration data under different inter-hole delay times are compared in

Table 6. Comparison of blast-induced vibration data under different inter-hole delay times.

Monitoring Point	Delay Time/ms	Distance from Blast Centre/m	PPV/(cm·s−1)
			Horizontal Radial Direction	Horizontal Tangential Direction	Vertical Direction
1	25	340	0.3981	0.4223	0.4752
	42	340	0.5131	0.4838	0.6386
2	25	370	0.2850	0.2917	0.4013
	42	370	0.3531	0.4095	0.4631
3	25	450	0.1923	0.2307	0.2321
	42	450	0.3378	0.2769	0.3007

. As shown in Table 6, when the inter-hole delay time is 42 ms, the maximum PPV values at the three monitoring points are 0.5131 cm/s in the horizontal radial direction, 0.4838 cm/s in the horizontal tangential direction, and 0.6386 cm/s in the vertical direction. When the inter-hole delay time is 25 ms, the corresponding maximum PPV values are 0.3981 cm/s, 0.4223 cm/s, and 0.4752 cm/s, respectively.

For both test schemes, the PPV values in the horizontal and vertical directions at monitoring point 1 are clearly higher than those at monitoring points 2 and 3. This is because the energy of blasting seismic waves decreases with increasing propagation distance. Under broadly similar blasting parameters and topographical and geological conditions, the PPV values at different monitoring points are lower for the 25 ms inter-hole delay time than for the original 42 ms delay scheme. The standards of many countries specify limit values for building vibrations to prevent damage to buildings caused by high vibration levels [34]. The principal frequencies of the seismic waves generated by blasting in this study primarily fall within the range of 10–30 Hz. According to China’s safety criteria for blasting vibrations, the PPV for residential buildings in this frequency range must not exceed 2–2.5 cm/s. All PPV recorded in this study did not surpass 2 cm/s, thus posing no adverse effects on residential buildings. In contrast, Germany’s safety criteria specify that the PPV for residential buildings in this frequency range should not exceed 0.5–1.5 cm/s. The original blasting scheme exhibited PPV exceeding 0.5 cm/s at monitoring point 1, which is detrimental to building stability. However, after adjusting the delay time, the PPV were all less than 0.5 cm/s, thereby reducing the impact of blasting vibrations on residential buildings. Variations in safety criteria across countries may be attributed to differences in building structures and materials. Research on this topic could be conducted in the future.

Using the PPV from the mine’s original blasting plan as a baseline, the vibration-reduction rates at various monitoring points were calculated. The results are shown in Figure 14. For blasting with delay time of 25 ms, the maximum and minimum vibration-reduction rates of PPV in the horizontal radial direction are 43.1% and 19.3%, respectively. The corresponding values are 28.8% and 12.7% in the horizontal tangential direction, and 25.6% and 13.3% in the vertical direction.

For the horizontal radial, horizontal tangential, and vertical directions, the average vibration-reduction rates are 28.3%, 19.4%, and 20.6%, respectively. These results indicate that an inter-hole delay time of 25 ms provides the best vibration-reduction performance. The vibration-reduction effect is most significant in the horizontal radial direction, followed by the vertical direction, and is relatively weaker in the horizontal tangential direction. The field test of open-pit deep-hole blasting confirms that the 25 ms inter-hole delay time provides good vibration reduction in all three directions. It also supports the feasibility of determining the optimal inter-hole delay time based on the dominant principal frequency.

The propagation of blast seismic waves is a complex, nonlinear process. The characteristics of blast seismic waves are influenced by various factors. This complexity far exceeds what can be described by an idealized sine function. This paper simplifies actual blast seismic waves into harmonic motion, which is an approximation and simplification. The primary purpose is to clearly illustrate the relationship between phase differences and the results of wave superposition. It explains the principle of vibration reduction from a theoretical perspective. However, due to the inherent complexity of blast seismic waves, it is difficult to achieve complete cancellation of wave amplitudes. The actual average vibration reduction rate achieved was only 28.3%. Future work needs to develop waveform models that can capture the intricacy of blast seismic waves. It should also determine how these waves superimpose, so that vibration-reduction effectiveness can be further enhanced.

It should be noted that the 25 ms delay time proposed in this paper is an optimized result obtained under the specific rock mass conditions and blasting parameters of this mining area. Therefore, on-site calibration and validation must be conducted before application at other mines.

6. Conclusions

This research examines the differences in blasting seismic wave characteristics between single-hole and group-hole blasting under open-pit deep-hole blasting. The dominant principal frequency of blasting seismic waves is defined. A method was then proposed for determining the optimal inter-hole delay time by combining the dominant principal frequency with the staggered-phase superposition theory. The main conclusions are as follows:

(1) The PPV of blasting seismic waves from both group-hole and single-hole blasting decreases with increasing distance from the blast centre. The attenuation rate of PPV in the vertical direction is higher than that in the horizontal directions. The statistical indicators of the principal frequencies of group-hole and single-hole blasting seismic waves differ only slightly. The principal frequencies approximately follow a normal distribution.

(2) Compared with maximum likelihood estimation and Bayesian analysis, the Bootstrap method gives narrower confidence intervals for the mean principal frequencies of blasting seismic waves. The dominant principal frequency of single-hole blasting seismic waves is 20 Hz. Based on the staggered-phase superposition theory, the optimal inter-hole delay time was calculated as 25 ms.

(3) Engineering application results demonstrate that the 25 ms delay time achieves good vibration-reduction effect. Compared with the original blasting scheme with an inter-hole delay time of 42 ms, blasting with a 25 ms inter-hole delay time gives average vibration-reduction rates of 28.3%, 19.4%, and 20.6% in the horizontal radial, horizontal tangential, and vertical directions, respectively. These research findings can serve as a reference for optimizing delay times and controlling blasting vibrations.