Statistical analysis of blast-induced vibration near an open pit mine

Abstract

Blast-induced vibration may be harmful to facilities in the vicinity of operating mines, mainly causing structural damage and human discomfort. This study presents an application of multivariate statistics to predict vibration levels regarding their potential to cause structural damage and human discomfort. An extensive seismic monitoring campaign was executed in a large open-pit iron ore mine, near a small village, to gather a dataset for a predictive multivariate analysis. Ten blasting events have produced a dataset of 158 valid measurements. Three classes of vibration peak velocity were adopted from legal standards, which later supported a cluster analysis. Then, it was possible to compare how much these two classification modalities respond to discriminant analysis. The next step was to carry out a principal component analysis (PCA) from the original database, and, comparatively, to plot both the scores concerning the classes derived from the vibration standard and those from the groups obtained from cluster analysis. PCA has considerably explained the data variability, while the three classes from cluster analysis resulted very similar to the corresponding ones from the vibration standards. The results have demonstrated that multivariate statistics may be applied to manage blasting-induced vibration and its deleterious effects with few adjustments and automation.

Key words
rock blasting; vibration; multivariate analysis; cluster analysis; discriminant analysis; principal component analysis

INTRODUCTION

Rock blasting is a unitary operation of critical importance in hard rock mining and should be designed to combine the best fragmentation (resulting in ideal particle size distribution, with appropriate fragment blockiness and avoiding both over-sized fragments and those fines from over-fracturing) and minimal cost, considering health, industrial hygiene and environmental restrictions (Aler et al. 1996). Among the negative blasting impacts on its surroundings are induced ground vibrations (Hudaverdi et al. 2007, 2011, Ak et al. 2009). The level of vibrations is influenced by two specific groups of parameters. First, the parameters inherent to the rock mass where the waves propagate: its physical properties, such as porosity, mechanical strength, the presence of joint sets, elasticity parameters (Young’s modulus, Poisson’s modulus, acoustic impedance). Obviously, these parameters are intrinsic to the mined region (NBR 9653:2005, 2005). The other group is concerned with the blasting plan, designed by the engineers, such as charge per delay, distance among blast holes and rows, bench height, drill hole diameter, stemming, delay between holes and rows ignition, sequence, and direction of initiation and subdrilling (Bhandari 1997).

The complexity of the phenomena involved in rock blasting makes the accurate modeling of its features challenging. In line with this statement, various attempts are found in the technical literature, seeking a predictive equation for blast-induced vibration (Fişne et al. 2010). Table I exemplifies this effort, with compiled cases by Kuzu (2008).

Table I
Regression equations for induced ground vibrations according to many authors (modified from Kuzu 2008).

Researchers	Empirical models	Researchers	Empirical models
Duvall & Petkof (1959)	v = k(R/Q1/2)-b	Kahriman (2004)	v = 0.34D-1.79
Langefors & Kihlstrom (1978)	v = k(Q/R2/3)b/2	Kahriman et al. (2006)	v = 0.561D-1.432
Arnbraseys & Hendron (1968)	v = k(R/Q1/3)-b	Rai & Singh (2004)	v = kR-b(Qmax)e-b
Nicholls et al. (1970)	v = 0.36D-1.63	Nicholson (2005)	v = 0.438D-1.52
BIS (1973) [IS 6922]	v = k(Q2/3/R)1.25	Rai et al. (2005)	Qmax = k(vD2)b
Siskind et al. (1980)	v = 0.828D-1.32	Ozer (2008) (sandstone)	v = 0.257D-1.03
Ghosh & Daemen (1983)	v = k(R/Q1 / 2 )-b e - α R	Ozer (2008) (shale)	v = 6.31D-1.9
Ghosh & Daemen (1983)	v = k(R/Q1/3)-be-αR	Ozer (2008) (limestone)	v = 3.02D-1.69
Roy (1991)	v = n + k(R/Q1/2)-1	Ak et al. (2009)	v = 1.367D-1.59
Roy (1991)	v = n + k(R/Q1/3)-1	Badal (2010)	v = 0.29D-1.296
CMRI (1993)	v = n + k(R/Q1/2)-1	Mesec et al. (2010)	v = 0.508D-1.37
Kahriman (2002)	v = 1.91D-1.13

As pointed out by Kumar and co-workers (Kumar et al. 2015), researchers generally perform site-specific experimental campaigns, as blast ground vibration is a concern. Seismographs are used in order to provide technical parameters, such as vertical, longitudinal, and transverse peak particle velocities. Their major resultant at a given instant, the so-called peak vector sum (PVS), is considered the main vibration parameter by many authors (Dehghani H & Ataee-Pour 2011). Along this line, Yilma (2016) performed a comparative study of the predictive equations of blast-induced vibrations and has suggested attenuation formulas. As Brazilian mining operation is concerned, Iramina et al. (2018) applied artificial neural network techniques and geomechanics parameters to model blast-induced vibration attenuation. As pointed by those authors, artificial neural network performed better, despite requiring larger datasets.

Database containing blast design parameters and particle velocity or acceleration are usually gathered by researchers, seeking to provide necessary information to support regression analysis that can predict the level of blast-induced vibration (Singh et al. 2008). This approach considers the raw measured blast data to predict ground vibrations.

The nomenclature of the parameters inserted in Table I is as follows: v — velocity of vibrations [mm/s]; k — a real multiplicative coefficient from the regression model; n — a real addictive coefficient from the regression model; Q — maximum charge of explosives per delay [kg]; R — radial distance [m]; D — scaled distance [m/kg² or m/kg³].

The importance of multivariate techniques applied to mining and geotechnical data has been confirmed by a number of applications in the area. Studies like those from Santos and coworkers (Santos et al. 2019), Sing and coworkers (Singh et al. 2015), Landim (2011), and Vezhapparambu and colleagues (Vezhapparambu et al. 2018) can be cited, just to name a few. This research is inserted in this context and aimed at studying ground vibration parameters using multivariate statistical analysis

Karadogan et al. (2014) made an interesting survey on legal norms concerning the limits of blast-induced vibrations in several countries, in order to guide their own technical study with a view to proposing similar standards for the Turkish mining sector. Nevertheless, for the present study, a combination of Brazilian legal vibration standard NBR 9653:2005 (Brazil 2005) stating vibration limit of 15.0 mm/s (for structure damage application), and the Australian one (Australia 2014) was adopted (vibration limit of 2.0 mm/s, for human comfort), since the latter is one of the few standards that adopts the human comfort as the criterion, in order to achieve a result that can embrace both structure damage and human comfort (Duvall & Fogelson 1962, Erten et al. 2009). From the adoption of these limits concerning the peak velocity of the particles under vibration, a multivariate statistical analysis was carried out to evaluate the discriminating power of some classic techniques of multivariate analysis of empirical data. The idea was to analyze the pertinence of future development of automated algorithms for predicting (and quantifying) the impacts resulting from blasting operations in the vicinity of the mine.

MATERIALS AND METHODS

The field investigations were performed at a large open pit iron mine located in the so-called Iron Quadrangle (Brazil) mineral province. The target mine is located near to a small village (Figure 1). The acquiring method was fully described by Silveira (2017) and Navarro Torres et al. (2018). A total of 20 seismographs (from Geosonics®, model SSU 3000 EZ+) were employed, in a campaign of 10 blast events, which has resulted in a large dataset of 158 valid measurements. 42 observations could not activate the trigger level of 0.5 mm/s. This trigger value was adopted because it was the least sensible practical limit for field survey in order to avoid recording events as car traffic or even pedestrians walking. The distance (D) between the vibration source and the monitored point was calculated by allocating UTM coordinates obtained by a Garmin GPS (model Map).

Figure 1
Representation of the region studied.

A combination of Brazilian legal vibration standard and the Australian corresponding one was adopted, since the latter is one of the few standards that adopts the human comfort as the criterion, in order to achieve a result that can embrace both structure damage and human comfort (Table II).

Table II
Summary of vibration limits adopted.

Standard	Vibration limit	Application
NBR 9653 (Brazil)	15.0 mm/s	Structure damage
Transport Noise Management (Australia)	2.0 mm/s	Human comfort

Due these criteria, it was created three different classes of vibration level: 15 < PVS, 2 < PVS < 15 and PVS < 2. These classes were also used for qualitative grouping of observations (k = 3).

The statistical analysis was initially performed at an exploratory level using Jamovi, a free and open statistical software package (The Jamovi Project 2019).

The cluster analysis of data was performed using Minitab® Version 19.2 (a proprietary system for statistical computation), which are widely used by statisticians. Cluster analysis of one dataset classifies the data into groups so that those within the same cluster share greater proximity of properties, when compared to data from the other groups (ultimately Euclidean distances between points belonging to a definite cluster are smaller than the distances between points of distinct clusters). Under this line of action, k-means clustering was employed. This clustering algorithm is used for partitioning a data set into a set of k pre-specified clusters. Using, intentionally, k = 3, two types of categories are obtained: those three ranges derived from vibration standards, and the three groups from cluster analysis. Then, it was possible to compare how much these two classification modalities respond to a discriminant analysis.

In sequence, a linear discriminant analysis was executed for the legal vibration limits. This technique is a dimensionality reduction tool applied on any homoscedastic dataset. It is an analysis with discriminant functions and is based on replacing original data with a linear combination of measurements that minimizes the variance and maximizes the distance between the means of the classes. Markedly, if standardization is carried out, z-scores are always homocedastic.

In the present case, the next methodological step was to perform a principal component analysis with the original database, and comparatively plot both the scores concerning the classes derived from the vibration standards and scores of the three groups from cluster analysis. The Jamovi statistical spreadsheet was used for this step.

Principal component analysis (PCA) is another very useful tool for multivariate data analysis based on reduction of dimensionality by an orthogonal coordinate transformation of a data set, generating a few orthogonal linearly uncorrelated variables called principal components, which capture most of the variability of those original data (Calabrese 2019). Therefore, the basic purpose of PCA is to transform a set of initial variables correlated with each other, into another set of uncorrelated (orthogonal) variables, called main components, which are appropriate linear combinations of the original variables. The components are calculated in descending order of importance, that is, the first main component explains the maximum possible variance of the original data, the second the maximum possible variance not yet explained by the first one, and so on. The last main component will be the one with the least contribution to explain the total variance of the original data. In a principal component analysis, all variables are treated in the same way, that is, there are no dependency relationships between variables, as occurs in regression analysis. The main objectives of this multivariate technique are to reduce the dimensionality of the data and to obtain interpretable combinations of the original variables.

The principal component analysis depends solely on the covariance matrix or the correlation matrix P of the original variables X1; ...; Xp. It is not necessary to assume the normality of the data for the application of the technique, as pointed out by Santos et al. (2019).

RESULTS AND DISCUSSION

Exploratory analysis

Table III presents the descriptive statistics of the variables used to compose the dataset used in this research. It was employed Jamovi spreadsheet for this analysis. In Table III and along the text, the symbols stand for the following parameters: D — distance [m]; Lv — wave longitudinal velocity [mm/s]; Tv — wave transverse velocity [mm/s]; Vv — wave vertical velocity [mm/s]; PVS — peak vector sum [mm/s]; and Q — explosive charge per delay [kg]. In turn, the subscript _st stands for “standardized variable”. The descriptive statistics of these data are showed in Table IV. In order to perform an exploratory analysis, a matrix plot of the data was done (Figure 2). This matrix provides a visual idea about the correlation between the variables and sometime called “draftsman’s plot”.

Figure 2
Matrix plot of the dataset (from Minitab).

Table III
Descriptive statistics (sample size: 158) showing the Shapiro-Wilk’s test for normality.

	D [m]	Lv [mm/s]	Tv [mm/s]	Vv [mm/s]	PVS [mm/s]	Q/delay [kg]
Mean	844.0	3.108	2.327	1.890	3.718	1495
Median	851.5	0.7300	0.7950	0.5100	0.9850	1500
Standard deviation	466.7	8.448	4.218	4.139	9.392	400.0
Minimum	52	0.06000	0.1300	0.06000	0.1900	800
Maximum	1955	65.53	34.04	31.69	74.93	2100
Shapiro-Wilk p	0.0010	< 0 .0001	< 0 .0001	< 0.0001	< 0.0001	< 0.0001
25th percentile	474.0	0.3800	0.3800	0.2500	0.5250	1250
50th percentile	851.5	0.7300	0.7950	0.5100	0.9850	1500
75th percentile	1203	2.083	2.030	1.635	2.652	1750

Table IV
Descriptive statistics for (dimensionless) standardized data (N = 158; mean = 0; variance = 1).

	D_st	Lv_st	Tv_st	Vv_st	PVS_st	Q/d_st
Median	0.016100	-0.28155	-0.36330	-0.33350	-0.29100	0.012700
Minimum	-1.6969	-0.36080	-0.52090	-0.44220	-0.37560	-1.7371
Maximum	2.3804	7.3889	7.5179	7.1998	7.5822	1.5125
Skewness	0.28519	5.7447	4.2095	4.9252	5.7140	-0.048856
Std. error skewness	0.19305	0.19305	0.19305	0.19305	0.19305	0.19305
Shapiro-Wilk p	0.0010	< .0001	< .0001	< .0001	< .0001	< .0001
25th percentile	-0.79270	-0.32300	-0.46170	-0.39630	-0.34000	-0.61230
50th percentile	0.016100	-0.28155	-0.36330	-0.33350	-0.29100	0.012700
75th percentile	0.76815	-0.12145	-0.070500	-0.061625	-0.11345	0.63760

A careful examination of the matrix plot has allowed inferring that there is some correlation with many of the studied variables. The corresponding boxplot of each measured (raw) variable would not be useful, due to scale discrepancy of variables (which are of very different magnitudes). In view of this, these parameters were standardized (transforming raw data to z-scores) and are shown in Figure 3. Z-scores of a generic variable x_i were obtained by the usual transformation:

Figure 3
Boxplots of standardized variables (from Jamovi).

$$X_{i\mathrm{\_}st}=z_i=\frac{(x_i-u_i)}{{\sigma }_i}$$ (1)

It is noteworthy that — at a first approach — some of these velocity measurements could be classified as anomalous, but in fact, they are not outliers (which are actually defined by statistical techniques), since the most discrepant values were those recorded at short distances from the blast faces and they were valid measurements obtained from the field campaign, which will be fully considered in the multivariate analysis.

K-means cluster analysis

In order to compare the legal standard classification adopted (which entails three ranges of vibration level) with the state of art multivariate statistical techniques, a k-means cluster analysis was performed, with three clusters, and standardized variables, as commented in this method explanation (item 2). The result of cluster analysis is displayed in Table V.

Table V
Final Partition.

	Number of observations	Within cluster sum of squares	Average distance from centroid	Maximum distance from centroid
Cluster1	3	41.860	3.612	4.742
Cluster2	131	240.026	1.235	2.309
Cluster3	24	85.866	1.742	4.168

Discriminant analysis

Initially a linear discriminant analysis was carried out for the legal vibration limits, using the standardized data because the original variables were not homoscedastic, as can be verified by the descriptive statistical analysis. Not surprisingly, in a second stage, the quadratic discriminant analysis of the original data proved to be more appropriate due to the lower elimination of measures resulting from excessive apparent correlation (as detected by the algorithm). The groups were: with PVS greater 15 mm/s (9 events), PVS between 2 mm/s and 15 mm/s (40 events), and PVS less than 2 mm/s (109 events). The correct classification proportion concerned legal vibration limits was 151/158 = 95.6 % for the quadratic discriminant function (against 89.2 % for linear function). After a cross validation procedure, in order to compensate for an optimistic apparent error rate of misclassified observations, the correct classification proportion was 147/158 = 93.3 %. Detailed results are presented in Table VI.

Table VI
Summary of classification from legal vibration limits.

	True Group (before cross validation)			True Group (after cross validation)
Put into Group	15 < PVS	2 < PVS < 15	PVS < 2	15 < PVS	2 < PVS < 15	PVS < 2
15 < PVS	9	1	0	8	4	0
2 < PVS < 15	0	38	5	1	35	5
PVS < 2	0	1	104	0	1	104
Total N	9	40	109	9	40	109
N correct	9	38	104	8	35	104
Proportion	1.000	0.950	0.954	0.889	0.875	0.954

The same process was applied for the cluster classification. The correct classifications proportion from cluster analysis was 156/158 = 98.7 %. Groups from cluster analysis were: group 1 (4 events), group 2 (117 events) and group 3 (37 events). Detailed results are presented, as shown in Table VII.

Table VII
Summary of classification from cluster analysis.

	True Group
Put into Group	1	2	3
1	4	0	0
2	0	115	0
3	0	2	37
Total N	4	117	37
N correct	4	115	37
Proportion	1.000	0.983	1.000

Principal component analysis

In order to reduce the number of dimensions of the dataset, preserving most of the original variability, a principal component analysis (PCA) was executed, as shown in Table VIII and Table IX. As already mentioned, the variables were standardized due to the high scale discrepancy of variables. Once this was done, a correlation matrix must be adopted, instead of a covariance matrix, as one of the PCA premises.

Table IX
Eigenvectors from PCA.

Variable	PC1	PC2	PC3	PC4	PC5
Distance [m]	-0.367	0.302	0.867	-0.134	-0.066
Longitudinal velocity (long_vel) [mm/s]	0.530	0.049	0.324	0.431	0.653
Transverse velocity (trans_vel) [mm/s]	0.537	0.037	0.090	-0.835	0.067
Vertical velocity (vert_vel) [mm/s]	0.543	0.062	0.199	0.313	-0.751
Charge per delay [kg]	0.033	0.949	-0.309	0.033	0.034

Table VIII
Eigenanalysis of correlation matrix from PCA.

	PC1	PC2	PC3	PC4	PC5
Eigenvalue	3.2191	1.0395	0.6253	0.0923	0.0239
Proportion	0.644	0.208	0.125	0.018	0.005
Cumulative	0.644	0.852	0.977	0.995	1.000

From these three techniques applied to the dataset, it was possible to reach the final results of multivariate assessment, as follows. Figure 4 shows the Mahalanobis distance for the observations, where some pseudo-outliers were detected, accordingly with the high variability explained before in the exploratory data analysis.

Figure 4
Outlier plot from principal component analysis (PCA).

Figure 5 displays the scree plot of eigenvalues of each component from PCA. Despite of criticism against its use (Yeomans & Golder 1982), the Guttman–Kaiser criterion for selecting the number of components was used because of its widespread employment in technical literature. Following the Guttman–Kaiser criterion, components were retained, which represent 85.2 % of the total variability of data. This cumulative proportion can be safely considered as a good representation of the total variability, plus a system dimension reduction from five to two.

Figure 5
Scree plot from principal component analysis (PCA).

Figure 6 shows a biplot of score from the first two components and the loading of all eigenvectors, in order to present the correlation between the data dispersion and the eigenvectors’ growth directions. As the distance increases, all velocities decrease, both ruled mostly by the first component, and the charge per delay grows in a direction ruled by the second component.

Figure 6
Biplot chart from principal component analysis (PCA).

Figure 7 was the score plot of the first two components, filtered by the legal vibration limits groups. There is a clear division in three groups, even though they were not defined by statistical techniques.

Figure 7
Score plot from principal component analysis (PCA) with legal vibration limits groups.

Figure 7 and Figure 8 show the same chart, but the former has the cluster group filter. The division of scores is clear too, and most important, it was almost identical to the legal vibration limit classification.

Figure 8
Score plot from principal component analysis (PCA) with cluster groups.

The parameters that determine the intensity and occurrence of the vibrations are vast and very heterogeneous, becoming extremally hard to predict and interpret without good statistical tools. In this mine-village context, depending on the regulation, the extrapolation of imposed limits can cause production stoppages, fines and affect the geotechnical structures such as slopes, stockpiles, waste piles and dams.

Once applied these statistical methods to the rock blasting data and to the mechanical properties of the rock mass, it was obtained a prediction model for the vibration intensity and its occurrence. Adding value through an anticipated and planned decision-making, reduction of production stoppage, and mitigation of eventual impacts on the geotechnical structures.

CONCLUSIONS

The impact of mining operations on communities located in its surroundings (which often arise from population settlement after the beginning of the mineral exploitation) has been the subject of many controversies among stakeholders and, consequently, studies aiming to alleviate this type of problem have been carried out. In this context, the adoption of computerized systems for the optimization of procedures, aiming at a lower impact due to blast vibrations, is extremely welcome.

The application of combined cluster, discriminant and principal component analysis allowed classifying this complex dataset under study, on a statistical basis, without losing the real application based on legal standards. In line with these results, one could think of those specialist systems and those based on artificial intelligence, just to mention two common techniques. In any case, these results of multivariate statistics may be important instruments for their implementation in the field, creating, for example, automatic systems for detecting occasional violations of preset limits of ground vibration.

Therefore, it will be possible to automate this type of analysis for forecasting of blast-induced vibration with respect to its harmful effects, once the k-means cluster analysis reaches an almost identical result compared to the legal vibration limit classes. One could implement, for instance, an algorithm incorporating statistical analysis and regression models into a decision tree, excluding the subjectivities inherent to the human factor. This methodological approach may be widely applied in the research area of induced ground vibrations from rock blasting.

ACKNOWLEDGMENTS

The authors are thankful to Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Fundação de Amparo à Pesquisa do Estado de Minas Gerais (FAPEMIG) and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) for their financial support. Special thanks to Gustavo Lopes, Vlademir Bastos and Péricles Luiz, due to their collaboration in seismic monitoring campaign.

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Publication Dates

• Publication in this collection
  14 Aug 2023
• Date of issue
  2023

History

• Received
  8 Jan 2021
• Accepted
  22 Mar 2021