# A Study of Blast Vibration Propagation Law under Negative Altitude Terrains

Authors: Jing Gao; Cong Huang; Xiaomin Huang; Jian-Jun Ren; Na Wang
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The propagation characteristics of blast waves and the prediction accuracy of blast vibration velocities along negative slopes are important because those can be used to guide engineering application and theoretical research. In this paper, the wave theory was applied to better understand the propagation mechanism of blast waves along negative slopes. Regression analysis was used for the Sadovsky and the CRSRI blast vibration velocity prediction models during on-site operations. The magnification of peak blast vibration velocity along a negative slope was introduced to determine the threshold altitude difference for the magnification effect to occur. Based on this parameter, the relative errors between the two prediction models were compared. The obtained results indicate that the superposition of incident and reflected blast waves on a negative slope creates the “slope effect” which locally amplifies the blast vibration velocity. The relative error of the CRSRI prediction model was as small as 17.53%, demonstrating a greater accuracy than Sadovsky’s prediction model. The magnification effect of a negative slope was observed at specific altitude differences and was more noticeable in the perpendicular direction. This paper creates a theoretical basis for studying the propagation mechanisms of blast vibration waves along negative slopes as well as predicting the blast vibration velocities.


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## Body

## 1. Introduction

Blasting operations usually induce a topographic change within the investigated area. These modifications decrease the accuracy of the Sadovsky formula which is commonly used for predicting the blast vibration velocity. The slope magnification effect caused by the propagation of blast waves over undulating terrain has become an essential part of studying the propagation of blast vibrations during rock slope excavations [1].Previous studies focused on the propagation mechanism of blast waves in rock slopes have laid the foundation for an accurate prediction of the blast vibration velocity. Chen et al. [2] suggested that the altitude magnification effect is caused by the “whipping effect,” as the main frequency of the blast wave load is equal to the natural vibration frequency of the bench. Guo et al. [3], Soltys et al. [4], and Man et al. [5] combined theoretical analyses with field tests and confirmed the “slope effect” for both negative and positive slopes, leading to an increase in the slope blast vibration velocity. Qiu [6] considered the rock slope as a multilayered environment and suggested that the blast waves would be superimposed by refraction and reflection after propagating to the interfaces of different layers. In consequence, this would create the “interface group effect” and lead to the altitude magnification effect.Several studies have been carried out to accurately predict the blast vibration velocity in terrains of varying altitudes. Liu et al. [7], Roy et al. [8], Elevli and Arpaz [9], and Torres et al. [10] applied Sadovsky’s formula with modifications in the altitude difference to predict the blast vibration velocity based on a large volume of field data. The conclusions showed that the modified formulas generated safer prediction results, but their solutions were not good supported by theoretical analysis since no normalization was implemented. Zhu et al. [11], Faradonbeh et al. [12], and Himanshu et al. [13] proposed a modified formula for predicting the blast vibration velocities for sloped terrains based on field data analysis and comparison experiment. They performed dimensional analyses on the altitude difference and obtained acceptable results when they applied the modified formula to practical situations. Tang [14] and Ainalis et al. [15] used dimensional analysis and deduced a formula for predicting blast vibration velocity based on the altitude difference. After applying the results to real-life scenarios, they concluded that the prediction error of the formula was as little as 10%. Li et al. [16] and Kumar et al. [17] studied the blast vibration of a shaft. Their results indicate that the altitude magnification effect had an elevation threshold above which the blast vibration velocity would be attenuated. Wu et al. [18] and Kim and Lee [19] implemented the linear regression approach to generate a predictive model for the blast vibration velocity in benched terrains based on numerical simulations and field tests. The results suggest that the correlation coefficient of the obtained formula was higher compared to the other formulas and could better reflect the influence of altitude difference on the blast vibration velocity. Liu et al. [20] used field tests as well as theoretical analyses and concluded that their modified prediction formula performed better than Sadovsky’s formula for terrains with a negative slope when the altitude difference was taken into consideration. Lei et al. [21] and Roy et al. [22] applied an already known altitude difference correction formula and regression fitting analysis approach and concluded that the slope coefficient (β) decreased during the initial stage but afterwards rapidly increased. Lu and Huetrulid [23], Agrawal and Mishra [24], and Hudaverdi [25] analyzed the issue from different theoretical angles and proposed a modified formula that was better theoretically adjusted and more comprehensive in predicting blast vibration velocity. Zhang [26] obtained a high-precision prediction formula for bench blast vibrations by analyzing the propagation mechanism of blast waves and by associating it with the elastic wave mechanics. Yang et al. [27] and Avellan et al. [28] studied the seismic topographic effect of blasting on the follow-wave slope and derived a model for predicting the blast vibration velocity of a follow-wave slope with higher differential elevations.The previously mentioned studies focused on the efficiency of the blast waves and the prediction of blast vibration velocities, but only some of them considered the negative slopes. Therefore, there is a need for further research. In this study, we investigated the propagation mechanism of blast waves along negative slopes using the elastic wave theory. Afterwards, we applied the regression analysis approach to two prediction models based on the altitude difference, and it was concluded that the prediction results of the CRSRI model were more accurate than those of the Sadovsky model. Finally, we introduced the magnification of peak vibration, so that the threshold altitude difference of the magnification effect of a negative slope could be quantified.


## 2. Analysis of the Mechanism of the Negative Slope Effect

### 2.1. Generation Mechanism of Seismic Waves

When explosives are detonated in rocks, a large amount of energy is instantaneously released. The generated energy is dissipated by shock wave overpressure, compressive stress waves, as well as seismic waves. This causes the formation of a crushing zone, fracture zone, and vibration zone spreading outward from the explosion center (Figure1). Within the vibration zone, seismic waves will cause elastic vibrations of the rock matrix, resulting in damage or even destruction of the matrix structure. Therefore, it is important to investigate the propagation mechanisms of blast waves in the vibration zone [29].Figure 1
Damaged zones caused by the explosionB-charge; R1-charging radius; R2-crushing zone; R3-fracture zone.


### 2.2. Basic Classification of Seismic Waves

The propagation of seismic waves in various media is affected by many factors, and the variety of seismic waves is extremely complex. Seismic waves propagating within a medium are called body waves, while those propagating near the surface are known as surface waves. The body waves can either be longitudinal or transverse. The longitudinal waves, whose propagation direction is aligned with the vibration direction of the medium, cause tensile and compressive damage to the medium, while the transverse waves, which propagate perpendicularly to the medium vibration direction, cause shear damage. Surface waves, on the other hand, are generally considered secondary waves which are generated by multiple reflections of the body waves from stratigraphic interfaces. The surface waves include Love waves and Rayleigh waves [30] (Figure 2).Figure 2
Classification of blast seismic waves.


### 2.3. Propagation Mechanism of Seismic Waves along a Negative Slope

According to the Fermat principle, the blast-generated seismic waves propagating in the negative slope direction will travel along the shortest path between the blast center and a measurement point, instead of a straight line [31] (Figure 3).Figure 3
Propagation mechanism of seismic waves along a negative slope.When the seismic wave propagates along a negative slope, the incident wave will be completely reflected from the slope, thus forming a reflective stretching area. Assuming that the slope is made of a homogeneous matrix, the propagation velocity of the incident wave can be expressed accordingly [32] as follows:(1)vit=A1ft.The propagation velocity of the reflected wave can be expressed as follows:(2)vrt=A1ft.By integrating the vector sum of the propagation velocities of the incident and the reflected wave, the displacement of slope particles can be calculated as follows:(3)Ut=∫0tvit−vrtdx,=2∫0tA1fxdx.In equations (1)–(3), A1 is the amplitude of the incident wave vibration velocity, vrt is the propagation velocity of the incident wave, and vrt is the propagation velocity of the reflected wave. It can be observed from equation (3) that a superposition of the incident and the reflected wave occurs when they travel along the negative slope direction, resulting in a local increase in vibration intensity.


## 2.1. Generation Mechanism of Seismic Waves

When explosives are detonated in rocks, a large amount of energy is instantaneously released. The generated energy is dissipated by shock wave overpressure, compressive stress waves, as well as seismic waves. This causes the formation of a crushing zone, fracture zone, and vibration zone spreading outward from the explosion center (Figure1). Within the vibration zone, seismic waves will cause elastic vibrations of the rock matrix, resulting in damage or even destruction of the matrix structure. Therefore, it is important to investigate the propagation mechanisms of blast waves in the vibration zone [29].Figure 1
Damaged zones caused by the explosionB-charge; R1-charging radius; R2-crushing zone; R3-fracture zone.


## 2.2. Basic Classification of Seismic Waves

The propagation of seismic waves in various media is affected by many factors, and the variety of seismic waves is extremely complex. Seismic waves propagating within a medium are called body waves, while those propagating near the surface are known as surface waves. The body waves can either be longitudinal or transverse. The longitudinal waves, whose propagation direction is aligned with the vibration direction of the medium, cause tensile and compressive damage to the medium, while the transverse waves, which propagate perpendicularly to the medium vibration direction, cause shear damage. Surface waves, on the other hand, are generally considered secondary waves which are generated by multiple reflections of the body waves from stratigraphic interfaces. The surface waves include Love waves and Rayleigh waves [30] (Figure 2).Figure 2
Classification of blast seismic waves.


## 2.3. Propagation Mechanism of Seismic Waves along a Negative Slope

According to the Fermat principle, the blast-generated seismic waves propagating in the negative slope direction will travel along the shortest path between the blast center and a measurement point, instead of a straight line [31] (Figure 3).Figure 3
Propagation mechanism of seismic waves along a negative slope.When the seismic wave propagates along a negative slope, the incident wave will be completely reflected from the slope, thus forming a reflective stretching area. Assuming that the slope is made of a homogeneous matrix, the propagation velocity of the incident wave can be expressed accordingly [32] as follows:(1)vit=A1ft.The propagation velocity of the reflected wave can be expressed as follows:(2)vrt=A1ft.By integrating the vector sum of the propagation velocities of the incident and the reflected wave, the displacement of slope particles can be calculated as follows:(3)Ut=∫0tvit−vrtdx,=2∫0tA1fxdx.In equations (1)–(3), A1 is the amplitude of the incident wave vibration velocity, vrt is the propagation velocity of the incident wave, and vrt is the propagation velocity of the reflected wave. It can be observed from equation (3) that a superposition of the incident and the reflected wave occurs when they travel along the negative slope direction, resulting in a local increase in vibration intensity.


## 3. Predictive Models for Negative Slopes and Their Coefficients

Sadovsky’s and CRSRI formulas are widely used in practical activities to predict blast vibration velocities. In this study, regression analysis is applied to these two common models to obtain the relative errors between the two prediction models.


### 3.1. Sadovsky’s Predictive Model Coefficients

(4)v=KQ3Rα,where v is the particle peak vibration velocity, m/s; Q represents the amount of explosive, kg; R is the distance from the observation point to the explosion center, m; K is a coefficient related to topographic and geological conditions; α is the attenuation coefficient.By applying the logarithm for both sides of equation (4), the following formula is obtained:(5)lgv=lgK+αlgQ3R.If we consider thaty=lgv, a = lgK, b = α, and x = lg Q3/R, (5) can be transformed into a single-variable linear function as follows:(6)y=a+bx.The least squares method can be used to calculate the coefficientsa and b as follows:(7)a=n∑i=1nxiyi−∑i=1nyi∑i=1nxin∑i=1nxi2−∑i=1nxi2,b=∑i=1nxiyi∑i=1nxi−∑i=1nyi∑i=1nxi2∑i=1nxi2−n∑i=1nxi2.


### 3.2. CRSRI Predictive Model Coefficients

(8)v=KQ3RαQ3Hβ,where H is the bench height; β is the influence coefficient of elevation difference; the other parameters have the same definition as in the previous equations.By applying the logarithm on both sides of equation (8), the following formula is obtained:(9)lgv=lgK+αlgQ3R+βlgQ3H.If we consider thatz = lgv, a = lgK, b = α, x = lg Q3/R, c = β, and y = lg(Q3/H), equation (9) can be transformed into a two-variable linear function as follows:(10)z=a+bx+cy.The least squares method can then be used to obtain the following equations:(11)∑i=1nxi2b+∑i=1nxiyic+∑i=1nxia=∑i=1nxizi,∑i=1nxiyib+∑i=1nyi2c+∑i=1nyia=∑i=1nyizi,∑i=1nxib+∑i=1nyic+na=∑i=1nzi,A=∑i=1nxi2∑i=1nxiyi∑i=1nxi∑i=1nxiyi∑i=1nyi2∑i=1nyi∑i=1nxi∑i=1nyin,B=∑i=1nxizi∑i=1nyizi∑i=1nziT,M=bcaT.WhenA ≠ 0, the only valid solution for M can be obtained by solving the system of linear equations as follows:(12)M=A−1B.


## 3.1. Sadovsky’s Predictive Model Coefficients

(4)v=KQ3Rα,where v is the particle peak vibration velocity, m/s; Q represents the amount of explosive, kg; R is the distance from the observation point to the explosion center, m; K is a coefficient related to topographic and geological conditions; α is the attenuation coefficient.By applying the logarithm for both sides of equation (4), the following formula is obtained:(5)lgv=lgK+αlgQ3R.If we consider thaty=lgv, a = lgK, b = α, and x = lg Q3/R, (5) can be transformed into a single-variable linear function as follows:(6)y=a+bx.The least squares method can be used to calculate the coefficientsa and b as follows:(7)a=n∑i=1nxiyi−∑i=1nyi∑i=1nxin∑i=1nxi2−∑i=1nxi2,b=∑i=1nxiyi∑i=1nxi−∑i=1nyi∑i=1nxi2∑i=1nxi2−n∑i=1nxi2.


## 3.2. CRSRI Predictive Model Coefficients

(8)v=KQ3RαQ3Hβ,where H is the bench height; β is the influence coefficient of elevation difference; the other parameters have the same definition as in the previous equations.By applying the logarithm on both sides of equation (8), the following formula is obtained:(9)lgv=lgK+αlgQ3R+βlgQ3H.If we consider thatz = lgv, a = lgK, b = α, x = lg Q3/R, c = β, and y = lg(Q3/H), equation (9) can be transformed into a two-variable linear function as follows:(10)z=a+bx+cy.The least squares method can then be used to obtain the following equations:(11)∑i=1nxi2b+∑i=1nxiyic+∑i=1nxia=∑i=1nxizi,∑i=1nxiyib+∑i=1nyi2c+∑i=1nyia=∑i=1nyizi,∑i=1nxib+∑i=1nyic+na=∑i=1nzi,A=∑i=1nxi2∑i=1nxiyi∑i=1nxi∑i=1nxiyi∑i=1nyi2∑i=1nyi∑i=1nxi∑i=1nyin,B=∑i=1nxizi∑i=1nyizi∑i=1nziT,M=bcaT.WhenA ≠ 0, the only valid solution for M can be obtained by solving the system of linear equations as follows:(12)M=A−1B.


## 4. Case Study

### 4.1. Geographical Location

The test site is in Guiyang City, Guizhou Province, China. The surroundings are complex, mainly comprising the low Zhongshan erosion residual hills and depressions. Overall, the terrain is relatively flat, and the ground elevation varies between 1330 m and 1350 m, that is, the maximum height difference is 20 m.Based on information from drilling logs and field observations, the site profile consists of soil, red clay, and plain fill. The thickness of the upper layer is 0.5–2 m, with an average thickness of 1.0 m. The bedrock is the first and the second section ofTsz1 + 2. The layers are consisted of argillaceous dolomite, claystones, tuffaceous claystones. The rocks are usually fractured. The site is located 60 m away from the Guizhou e-commerce industrial park to the east, 65 m away from a small-sized pig shed to the south, 72 m away from a single residential house, and 113 m away from the Binyang Road to the southeast, in proximity to residential areas to the west, and 331 m away from residential houses to the northwest. There are industrial parks, roads, high-intensity residential housing areas, and other similar elements located around the blasting area (Figure 4) [33].Figure 4
Blasting area surroundings.


### 4.2. Blasting Procedures

According to the on-site geological conditions and blasting requirements, this test adopts a multirow deep-hole blasting method, a hole-by-hole initiation technology, and the relevant [34, 35]. Rock powdery emulsion explosives and bulk ammonium nitrate explosives are used in the field, and the specific blasting parameters are shown in Tables 1 and 2.Table 1
Layout parameters of the deep-hole blasting.
Rock typeDensity (t/m3)Aperture mmHole arrangementPiercing modeBench heightH (m)Dolomite2.890TriangleVertical6∼8Table 2
Parameters used for the deep-hole bench blasting procedure.
Bench heightH (m)Resistance linew1 (m)Depth Δh (m)Pitch rowa (m)Array pitchb (m)Loaded lengthL1 (m)Unit consumption (kg/m3)Packing height (m)One charge of explosivesQ (kg)62.50.52.522.230.35≥3.010.572.80.52.82.53.4117.283.00.53.02.54.3721This design points out that it is allowed to use the above parameters in the indicated area only if the maximum single dose of explosives is less than 21 kg. The specific delay design is shown in Table3.Table 3
Delay combination scheme of hole-by-hole initiation technology.
Bench heightH (m)Pitch rowa (m)Delay in the hole (ms)Array pitchb (m)Delay between the rows (ms)Rock properties62.520230The argillaceous dolomite has developed joints and cracks; the rock mass is brittle; it is primarily weathered when crushed.72.8252.54083.0252.545In multirow deep-hole blasting, the triangle hole arrangement is selected on the step flat plate, as shown in Figure5.Figure 5
Schematic diagram of the shot hole layout and the delay settings.The initiation sequence is illustrated in Figure5. The numbers in the figure represent the delay time. Based on the delay network, the movement and the throwing direction of the explosive pile are indicated in Figure 6.Figure 6
Schematic diagram of the throwing and the moving direction of the explosion.The network has a parallel connection. It registers the electronic detonator, as well as the number of the main line of the network. The electronic detonator clip of each hole is connected to the main line and is firmly attached to it. After the registration is completed, the detonation network extension is pressed. The time difference can be set manually, and a special detonator is used for charging and detonation. The schematic diagram of the connection between the electronic detonators is illustrated in Figure7.Figure 7
Schematic diagram of the electronic detonator connection.


### 4.3. Field Monitoring

TC-4850 and NUBOX-8016 blast vibration meters were used during on-site operations. Both instruments can collect data from tangential, vertical, and radial channels (Figure8).Figure 8
Field monitoring system.Two rounds of data collection were carried out when the test was performed. Five measurement points were selected for each round, and ten field data records were obtained, as shown in Table4.Table 4
Blasting parameters and monitoring data.
Data collectionMeasurement pointDistanceMax. charge weight per delay,Q (kg)PPV,v (cm·s−1)Horizontal distance from blasting center,R (m)Vertical distance from blasting center,H (m)vvvtvrGroup 1160−4140.6890.3940.543270−6120.3830.2330.260380−8100.2560.2370.2524150−8.5120.3210.2000.2075155−12100.1270.0950.092Group 2160−4140.6360.3910.435270−6120.3730.2490.245380−8100.3150.2240.2134150−8.5120.4070.2480.2175155−12100.2870.1260.127


### 4.4. Analysis of Blast Vibration Monitoring Results

Group 1 experimental data were selected from Table4 to plot the vertical, tangential, and radial peak velocities according to the distance from the blast center (Figure 9).Figure 9
Peak vibration velocity vs. distance from the blast center.It can be observed from Figure9 that for the studied terrain with a negative slope, the peak velocities of blast vibrations in each of the three directions generally decreased with the increase in the distance from the blast center. However, when the distance from the blast center was equal to 150 m and 80 m, the peak velocities of blast vibration showed local magnification in the vertical and tangential direction, respectively. The magnification effect in the vertical direction was bigger than that in the tangential direction.


## 4.1. Geographical Location

The test site is in Guiyang City, Guizhou Province, China. The surroundings are complex, mainly comprising the low Zhongshan erosion residual hills and depressions. Overall, the terrain is relatively flat, and the ground elevation varies between 1330 m and 1350 m, that is, the maximum height difference is 20 m.Based on information from drilling logs and field observations, the site profile consists of soil, red clay, and plain fill. The thickness of the upper layer is 0.5–2 m, with an average thickness of 1.0 m. The bedrock is the first and the second section ofTsz1 + 2. The layers are consisted of argillaceous dolomite, claystones, tuffaceous claystones. The rocks are usually fractured. The site is located 60 m away from the Guizhou e-commerce industrial park to the east, 65 m away from a small-sized pig shed to the south, 72 m away from a single residential house, and 113 m away from the Binyang Road to the southeast, in proximity to residential areas to the west, and 331 m away from residential houses to the northwest. There are industrial parks, roads, high-intensity residential housing areas, and other similar elements located around the blasting area (Figure 4) [33].Figure 4
Blasting area surroundings.


## 4.2. Blasting Procedures

According to the on-site geological conditions and blasting requirements, this test adopts a multirow deep-hole blasting method, a hole-by-hole initiation technology, and the relevant [34, 35]. Rock powdery emulsion explosives and bulk ammonium nitrate explosives are used in the field, and the specific blasting parameters are shown in Tables 1 and 2.Table 1
Layout parameters of the deep-hole blasting.
Rock typeDensity (t/m3)Aperture mmHole arrangementPiercing modeBench heightH (m)Dolomite2.890TriangleVertical6∼8Table 2
Parameters used for the deep-hole bench blasting procedure.
Bench heightH (m)Resistance linew1 (m)Depth Δh (m)Pitch rowa (m)Array pitchb (m)Loaded lengthL1 (m)Unit consumption (kg/m3)Packing height (m)One charge of explosivesQ (kg)62.50.52.522.230.35≥3.010.572.80.52.82.53.4117.283.00.53.02.54.3721This design points out that it is allowed to use the above parameters in the indicated area only if the maximum single dose of explosives is less than 21 kg. The specific delay design is shown in Table3.Table 3
Delay combination scheme of hole-by-hole initiation technology.
Bench heightH (m)Pitch rowa (m)Delay in the hole (ms)Array pitchb (m)Delay between the rows (ms)Rock properties62.520230The argillaceous dolomite has developed joints and cracks; the rock mass is brittle; it is primarily weathered when crushed.72.8252.54083.0252.545In multirow deep-hole blasting, the triangle hole arrangement is selected on the step flat plate, as shown in Figure5.Figure 5
Schematic diagram of the shot hole layout and the delay settings.The initiation sequence is illustrated in Figure5. The numbers in the figure represent the delay time. Based on the delay network, the movement and the throwing direction of the explosive pile are indicated in Figure 6.Figure 6
Schematic diagram of the throwing and the moving direction of the explosion.The network has a parallel connection. It registers the electronic detonator, as well as the number of the main line of the network. The electronic detonator clip of each hole is connected to the main line and is firmly attached to it. After the registration is completed, the detonation network extension is pressed. The time difference can be set manually, and a special detonator is used for charging and detonation. The schematic diagram of the connection between the electronic detonators is illustrated in Figure7.Figure 7
Schematic diagram of the electronic detonator connection.


## 4.3. Field Monitoring

TC-4850 and NUBOX-8016 blast vibration meters were used during on-site operations. Both instruments can collect data from tangential, vertical, and radial channels (Figure8).Figure 8
Field monitoring system.Two rounds of data collection were carried out when the test was performed. Five measurement points were selected for each round, and ten field data records were obtained, as shown in Table4.Table 4
Blasting parameters and monitoring data.
Data collectionMeasurement pointDistanceMax. charge weight per delay,Q (kg)PPV,v (cm·s−1)Horizontal distance from blasting center,R (m)Vertical distance from blasting center,H (m)vvvtvrGroup 1160−4140.6890.3940.543270−6120.3830.2330.260380−8100.2560.2370.2524150−8.5120.3210.2000.2075155−12100.1270.0950.092Group 2160−4140.6360.3910.435270−6120.3730.2490.245380−8100.3150.2240.2134150−8.5120.4070.2480.2175155−12100.2870.1260.127


## 4.4. Analysis of Blast Vibration Monitoring Results

Group 1 experimental data were selected from Table4 to plot the vertical, tangential, and radial peak velocities according to the distance from the blast center (Figure 9).Figure 9
Peak vibration velocity vs. distance from the blast center.It can be observed from Figure9 that for the studied terrain with a negative slope, the peak velocities of blast vibrations in each of the three directions generally decreased with the increase in the distance from the blast center. However, when the distance from the blast center was equal to 150 m and 80 m, the peak velocities of blast vibration showed local magnification in the vertical and tangential direction, respectively. The magnification effect in the vertical direction was bigger than that in the tangential direction.


## 5. Analysis of the Negative Slope Effect on Blast Vibrations

The negative slope magnification effect on blast vibration velocity was analyzed in detail by considering that the magnification effect in the vertical direction was more pronounced. The first mid-vertical monitoring data were selected for the linear regression analysis of the Sadovsky and the CRSRI formulas.


### 5.1. Regression Analysis of Sadovsky’s Predictive Model

The results of the linear regression approach for Sadovsky’s formula are illustrated in Figure10.Figure 10
Regression analysis based on Sadovsky’s predictive model.By including the regression parameters into equations (5) and (6), equation (4) can be expressed as(13)v=15.9945Q3H1.05598.


### 5.2. Regression Analysis of the CRSRI Predictive Model

To obtain a more accurate blast vibration prediction model, the linear regression approach was applied to the CRSRI formula, and the resulting model coefficients are illustrated in Table5.Table 5
Multiple regression parameters for the CRSRI predictive model.
Interceptlg{{Q1/3}/R}lg{{Q1/3}/H}AnalysisValueStandard errorValueStandard errorValueStandard errorAdjustedR2lgvv−0.261 50.4851−0.357 10.407 41.64580.420 80.9179By including the parameters from Table4 into equations (9) and (10), equation (8) can be expressed as(14)v=0.5476×Q3H−0.3571Q3H1.6458.The experimental data from Table4 are included in equations (13) and (16) to obtain predicted values by using the Sadovsky and the CRSRI formulas. The relative errors of the two formulas were calculated as(15)Relative  error=Predicted value−Measured valueMeasured value×100%and are illustrated in Table 6.Table 6
Relative errors of the predicted velocities.
Data collectionMeasurement pointPPV,v (cm·s−1)Predicted velocity from Sadovsky’s formula, (cm/s)Predicted velocity from the CRSRI formula, (cm/s)Relative error of Sadovsky’s formula, (%)Relative error of the CRSRI formula, (%)vvvtvrGroup110.689 00.394 00.543 00.536 70.749 722.118.8120.383 00.233 00.260 00.432 00.380 412.780.6830.256 00.237 00.252 00.351 80.229 837.4410.2440.321 00.200 00.207 00.193 20.281 539.8212.3150.127 00.095 00.092 00.175 00.149 337.7917.56Group 210.636 00.391 00.435 00.536 70.749 715.6217.8820.373 00.249 00.245 00.432 00.380 415.811.9830.314 70.224 00.213 00.351 80.229 811.8026.9840.407 00.248 00.217 00.193 20.281 552.5430.8450.287 00.126 00.127 00.175 00.149 339.0347.98Average error28.4717.53According to Figure10 and Table 6, the R2 value for Sadovsky’s prediction model is 0.526, with the average relative error of the predicted velocity equal to 28.47%. This indicates a poor correlation and large errors for this prediction model. In contrast, according to Tables 3 and 4, the R2 value for the CRSRI prediction model is 0.918, and the average relative error is equal to 17.53%, which means that the CRSRI prediction model of blast vibration is more accurate for negative slopes.


### 5.3. Analysis of Peak Velocity Magnification for the Negative Slope

The peak velocity magnification parameter (ε) was used to calculate the threshold slope difference for the negative slope magnification effect. Since the values of vm and vd are always positive, it is observed from equation (16) that when vm > vd, then values where ε > 0 result in the slope magnification effect. The formula for ε is expressed as [36](16)ε=vm−vdvd×100%,where vm is the peak velocity calculated by the CRSRI prediction model, and vd is the peak velocity calculated by the Sadovsky’s prediction model. The peak blast vibration velocities predicted using the Sadovsky and the CRSRI models listed in Table 4 were introduced into equation (16) to calculate the peak velocity magnification at the measurement points (Table 7). This allowed establishing the relationship between the peak vertical velocity magnification and the altitude difference (Figure 11). For visualization purposes, all negative altitude differences were plotted as absolute values.Table 7
Peak vertical vibration velocity magnification.
Measurement pointAltitude difference, (m)vdvmε1−40.749 70.536 70.396 972−60.380 40.432 0−0.119 383−80.229 80.351 8−0.346 914−8.50.281 50.193 20.457 305−120.149 30.175 0−0.146 79Figure 11
Variation in peak velocity magnification with the altitude difference.As seen in Figure11, except for the edge point (where the altitude difference was −4 m, and the peak velocity magnification was 0.3970 > 0), only when the altitude difference was −8.5 m, the peak velocity magnification was 0.4573 > 0, and the peak velocity magnification corresponding to the other altitude differences was smaller than 0 in all cases. This means that the altitude magnification in the vertical direction occurred solely when the altitude difference reached −8.5 m, that is, the altitude threshold for the altitude magnification effect was equal to −8.5 m. According to Figure 9, the magnification effect of the negative slope was produced only for certain altitude differences, and it was more pronounced in the vertical direction.


## 5.1. Regression Analysis of Sadovsky’s Predictive Model

The results of the linear regression approach for Sadovsky’s formula are illustrated in Figure10.Figure 10
Regression analysis based on Sadovsky’s predictive model.By including the regression parameters into equations (5) and (6), equation (4) can be expressed as(13)v=15.9945Q3H1.05598.


## 5.2. Regression Analysis of the CRSRI Predictive Model

To obtain a more accurate blast vibration prediction model, the linear regression approach was applied to the CRSRI formula, and the resulting model coefficients are illustrated in Table5.Table 5
Multiple regression parameters for the CRSRI predictive model.
Interceptlg{{Q1/3}/R}lg{{Q1/3}/H}AnalysisValueStandard errorValueStandard errorValueStandard errorAdjustedR2lgvv−0.261 50.4851−0.357 10.407 41.64580.420 80.9179By including the parameters from Table4 into equations (9) and (10), equation (8) can be expressed as(14)v=0.5476×Q3H−0.3571Q3H1.6458.The experimental data from Table4 are included in equations (13) and (16) to obtain predicted values by using the Sadovsky and the CRSRI formulas. The relative errors of the two formulas were calculated as(15)Relative  error=Predicted value−Measured valueMeasured value×100%and are illustrated in Table 6.Table 6
Relative errors of the predicted velocities.
Data collectionMeasurement pointPPV,v (cm·s−1)Predicted velocity from Sadovsky’s formula, (cm/s)Predicted velocity from the CRSRI formula, (cm/s)Relative error of Sadovsky’s formula, (%)Relative error of the CRSRI formula, (%)vvvtvrGroup110.689 00.394 00.543 00.536 70.749 722.118.8120.383 00.233 00.260 00.432 00.380 412.780.6830.256 00.237 00.252 00.351 80.229 837.4410.2440.321 00.200 00.207 00.193 20.281 539.8212.3150.127 00.095 00.092 00.175 00.149 337.7917.56Group 210.636 00.391 00.435 00.536 70.749 715.6217.8820.373 00.249 00.245 00.432 00.380 415.811.9830.314 70.224 00.213 00.351 80.229 811.8026.9840.407 00.248 00.217 00.193 20.281 552.5430.8450.287 00.126 00.127 00.175 00.149 339.0347.98Average error28.4717.53According to Figure10 and Table 6, the R2 value for Sadovsky’s prediction model is 0.526, with the average relative error of the predicted velocity equal to 28.47%. This indicates a poor correlation and large errors for this prediction model. In contrast, according to Tables 3 and 4, the R2 value for the CRSRI prediction model is 0.918, and the average relative error is equal to 17.53%, which means that the CRSRI prediction model of blast vibration is more accurate for negative slopes.


## 5.3. Analysis of Peak Velocity Magnification for the Negative Slope

The peak velocity magnification parameter (ε) was used to calculate the threshold slope difference for the negative slope magnification effect. Since the values of vm and vd are always positive, it is observed from equation (16) that when vm > vd, then values where ε > 0 result in the slope magnification effect. The formula for ε is expressed as [36](16)ε=vm−vdvd×100%,where vm is the peak velocity calculated by the CRSRI prediction model, and vd is the peak velocity calculated by the Sadovsky’s prediction model. The peak blast vibration velocities predicted using the Sadovsky and the CRSRI models listed in Table 4 were introduced into equation (16) to calculate the peak velocity magnification at the measurement points (Table 7). This allowed establishing the relationship between the peak vertical velocity magnification and the altitude difference (Figure 11). For visualization purposes, all negative altitude differences were plotted as absolute values.Table 7
Peak vertical vibration velocity magnification.
Measurement pointAltitude difference, (m)vdvmε1−40.749 70.536 70.396 972−60.380 40.432 0−0.119 383−80.229 80.351 8−0.346 914−8.50.281 50.193 20.457 305−120.149 30.175 0−0.146 79Figure 11
Variation in peak velocity magnification with the altitude difference.As seen in Figure11, except for the edge point (where the altitude difference was −4 m, and the peak velocity magnification was 0.3970 > 0), only when the altitude difference was −8.5 m, the peak velocity magnification was 0.4573 > 0, and the peak velocity magnification corresponding to the other altitude differences was smaller than 0 in all cases. This means that the altitude magnification in the vertical direction occurred solely when the altitude difference reached −8.5 m, that is, the altitude threshold for the altitude magnification effect was equal to −8.5 m. According to Figure 9, the magnification effect of the negative slope was produced only for certain altitude differences, and it was more pronounced in the vertical direction.


## 6. Conclusions

This study applied the wave theory to analyze the propagation mechanism of blast waves along negative slopes. The linear regression approach was used for on-site data for the Sadovsky and the CRSRI blast vibration velocity prediction models. To compare their relative errors, the magnification of peak blast vibration velocity along a negative slope was introduced to determine the threshold altitude difference for the magnification effect to occur. The conclusions are as follows:(1)
In terrains with a negative slope, the superposition of incident and reflected blast vibration waves on the slope results in the local amplification of blast vibration velocity.(2)
The influence of the negative slope on the peak blast vibration velocity should be acknowledged. The predictions of the peak blast vibration velocity using the CRSRI formula are more accurate when compared to the other formulas.(3)
The magnification effect of the negative slope appears only for certain altitude differences above a threshold and is more obvious for the vertical vibration velocity.



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*Source: 4289057-2022-03-10.xml*