A method for predicting blast-induced ground vibration based on Mamdani Fuzzy Inference System

Abstract

Blast-induced ground vibration is highly possible to result in serious losses such as destroyed buildings. The crucial parameter of the mentioned vibration is peak particle velocity (PPV). Many equations have been developed to predict PPV, however, worse performance has been reported by multiple literatures. This paper developed a method for predicting PPV based on Mamdani Fuzzy Inference System. Firstly, Minimum Redundancy Maximum Relevance was employed to identify the blasting design parameters which significantly contribute to the PPV induced by blasting. Secondly, K-means method was applied to determine the value ranges of the selected parameters. The selected parameters and corresponding value ranges were combined to input into Mamdani Fuzzy Inference System for obtaining predicted PPV. Totally, 280 samples were collected from a blasting site. 260 out of them were used to train the proposed method and 20 were assigned for test. The proposed method was tested in the comparison with empirical equation USBM, multiple linear regression analysis, pure Mamdani Fuzzy Inference System in terms of the difference between predicted PPV and measured PPV, coefficient of correlation, root-mean-square error, and mean absolute error. The results from that showed that the proposed method has the better performance in PPV prediction.

Keywords

Blasting peak particle velocity parameter selection k-means method Mamdani Fuzzy Inference System

1 Introduction

Blasting operation is widely used to be the most economical technique for rock mass fragment in mine works. However, many studies have revealed that only 20% –30% of the amount of total explosive energy is converted into mechanical energy for fragmenting [1, 2]. The majority of explosive energy generates side effects such as high temperature, noise, flying rocks, and air overpressure. Of them, blast-induced ground vibration is highly possible to result in serious losses through creating severe resonance for buildings in vicinity [3]. Generally, blast-induced ground vibration is under the influence of, besides rock mass properties, explosive properties and blasting geometry such as maximum charge used per delay (W), burden (B), spacing (S), stemming (T), blast-hole depth (H), blast-hole diameter (D_h), powder factor (PF), where burden means the distance from borehole to free face and stemming means the difference of the total length of borehole and the length of explosive [4, 5]. Blast-induced ground vibration can be evaluated from various dimensions including acceleration, frequency, and peak particle velocity (PPV). Amongst them, PPV, meaning the velocity of motion of a particle on or in the ground induced by the passing of the blast vibration waves, is increasingly considered as the crucial parameter for measuring vibration intensity [6, 7].

Over the past decades, many empirical methods have been developed for predicting PPV such as the United States Bureau of Mines (USBM) equation [8], Langefors–Kihlstrom model [9], Bureau of Indian Standards [10]. The common feature of them is that only focusing on the maximum charge per delay and the distance from the monitoring point to the blasting face (D). Some of them are listed in Table 1.

Table 1
Different empirical methods for PPV prediction

Developers Equation Note

USMB $PPV = k {(D / \sqrt{W})}^{- b}$ W:maximum charge per delay (kg)

Langefors–Kihlstrom $PPV = k {(\sqrt{W / D^{2 / 3}})}^{b}$ D: the distance between blast face to monitoring point (m)

Bureau of Indian Standards PPV = k (W/D^2/3) ^b

Central Mining Research Institute of India [11] $PPV = n + k {(D / \sqrt{W})}^{- 1}$ k, b, and n are constants having different values

Ambraseys and Hendron [12] $PPV = k {(D / \sqrt[3]{W})}^{- b}$

Developers	Equation	Note
USMB	$PPV = k {(D / \sqrt{W})}^{- b}$	W:maximum charge per delay (kg)
Langefors–Kihlstrom	$PPV = k {(\sqrt{W / D^{2 / 3}})}^{b}$	D: the distance between blast face to monitoring point (m)
Bureau of Indian Standards	PPV = k (W/D^2/3) ^b
Central Mining Research Institute of India [11]	$PPV = n + k {(D / \sqrt{W})}^{- 1}$	k, b, and n are constants having different values
Ambraseys and Hendron [12]	$PPV = k {(D / \sqrt[3]{W})}^{- b}$

Recently, more and more literatures claim that the contributory factors like burden and stemming also contribute to PPV [13, 14], which results in a direct conclusion that it is hard to obtain a precise PPV using only W and D. To comprehensively consider possible contributors, some artificial intelligence techniques have been developed. Some researchers invented an Artificial Neural Network (ANN) method with W, D, T, and the number of hole-rows as inputs [15]. Some scholars developed a hybrid method with Fuzzy C-means Clustering (FCC) and Support Vector Regression (SVR) as components. In which FCC divided the data with B, S, T, W, D, and the number of holes per delay into several clusters, then SVR predicted PPV for each cluster [16]. Some experts proposed to use Principal Component Analysis (PCA) to simplify the inputs of the Support Vector Machine (SVM) which was designed to predict PPV. Inputs included D, W, B, S, H, total charge, number of delays, and horizontal distance [17]. A study utilized Feature Selection (FS) technique to reduce the data dimensionality from six to be five, including S, D, PF, W, and H. These five outputs were inputted into Random Forest (RF) model and a Bayesian network to predict PPV [18].

Additionally, some papers have explored the application of fuzzy tools in PPV prediction [19, 20]. Amongst them, the most widely used fuzzy tool is Mamdani Fuzzy Inference System (MFIS). Compared with other artificial intelligence tools, MFIS has advantages in intuitive expression, widespread acceptance, and human cognition [21, 22]. Some studies have implied that MFIS may work better, in the case that the number of inputs is not over a limit, in the comparison with other machine learning approaches [23, 24]. Currently, there are not many published articles regarding PPV prediction with MFIS. The existing literatures regarding PPV prediction with MFIS have failed to deal with parameter selection and membership function establishment well. Parameter selection is the process for identifying the significant variables contributing to PPV. This process is capable of, in an acceptable manner, excluding irrelevant variables and keeping useful variables to participate in the PPV determination. Each selected variable has to be provided with a membership function which is used as the input to MFIS. Membership functions have to be determined based on sample data to improve the accuracy of predicted PPV, which means the value ranges of each variable must be consistent to the information contained by data. However, previous studies have defaulted membership functions as the ones defined by MFIS. In the case, they did not sufficiently integrate data into membership functions.

The present study attempted to overcome above two drawbacks. Minimum Redundancy Maximum Relevance (MRMR) [25, 26] was employed to conduct parameter selection. K-means method (KMM) [27] was applied to identify the value ranges of each selected variable. Meanwhile, the results from KMM are inputted into MRMR to provide the later with data clusters. Then, the selected variables and value ranges are connected to form the membership functions which are inputted into MFIS for outputting the value of predicted PPV. The flowchart of the proposed method is shown in Fig. 1.

The remainder of the present study is organized as follows: Section 2 elaborates the proposed method. Section 3 makes a case study. Section 4 conducts a comparison study with three methods including Mamdani Fuzzy Inference without parameter selection and membership functions analysis, USBM equation, and multiple linear regression analysis (MLRA). Section 5 evaluates the proposed method. Section 6 concludes the paper.

Fig. 1

The flowchart of the proposed method.

2 The proposed method

2.1 K-means method

KMM is an iterative clustering algorithm. It aims at automatically putting a number of data points from a dataset with multiple dimensions into K clusters so that the variation of data in the same cluster is minimum. This algorithm complies with the following loop: 1) arbitrarily assign an original centroid for each of K clusters; 2) associate each data point belonging to the given set to the centroid which has shortest distance to it. Generally, the distance is considered as Euclidean distance, then sum the distances for all clusters; 3) calculate the mean of the data points in a cluster to be a new centroid for each cluster, then repeat step 2); 4) if the sum in step 3) is smaller, repeat above steps until no new centroids are generated. The outputted centroids can be mathematically represented as Equation (1), where C means centroid; C_i means the i^th centroid in single loop; X means the data within a cluster with C_i as centroid. $\overset{a r g m i n}{C} = \sum_{i = 1}^{K} \sum_{X \in C_{i}} X - C_{i}^{2}$ (1)

In the present study, each dimension of data is independently divided into K states. For example, “W” can be assigned to be 3 states with labels of Low, Medium, High, respectively. Each state has an exclusively specific range of values.

2.2 Minimum Redundancy Maximum Relevance

MRMR, as the name suggests, is comprised by two components, namely minimum redundancy (R) and maximum relevance (Dr), as shown by Equations (3) and (4), respectively, where X and Y are a pair of variables, x and y mean the possible values of X and Y, respectively; p (x) and p (y) mean the proportion of the samples having any value of x and y, respectively, to total samples; p (x, y) means the proportion of the samples having any pair of values of x and y to total samples; |S| denotes the number of features of selected feature set; c means the target feature; X_i and X_j mean the i^th and the j^th feature (i ≠ j), respectively. MRMR scores each feature with an indicator named Mutual Information Difference (MID), as shown in Equation (5). The features with higher scores are recommended to be kept. $I (X, Y) = \iint p (x, y) \log \frac{p (x, y)}{p (x) p (y)}$ (2) $R = (\frac{1}{{| S |}^{2}} \sum I (X_{i}, X_{j}))$ (3) $Dr = (\frac{1}{| S |} \sum I (X_{i}, c))$ (4) $MID = Dr - R$ (5)

2.3 Mamdani Fuzzy Inference System

Outputted membership functions are inputted into MFIS. MFIS was initially proposed to control a steam engine and boiler combination. In sequence, MFIS consists of three components: fuzzification, inference rules, and defuzzification, as shown in Fig. 2.

Fig. 2

The diagram of process of Mamdani Fuzzy Inference.

Fuzzification utilizes membership functions to convert the crisp values of both inputs and outputs into linguistic fuzzy variables. Applicable membership functions contain Gaussian, Trapezoid, Triangle, and Sigmoid, etc. [28]. Membership functions can be recorded in several fuzzy sets. For example, term “temperature” can be fuzzificated into “Low” (L), “Medium” (M), and “High” (H). The upper and lower limits for all fuzzy sets can be quantitatively defined by the value ranges outputted by KMM. Inference rules connect the input and output variables by a set of IF-THEN rules which can be understood as that if antecedent is given, then accordingly the consequent is defined. For example, there are two rules: “If x₁ is A₁ and x₂ is B₁, then y is C₁; If x₁ is A₂ and x₂ is B₂, then y is C₂, where A₁ and A₂, as well as B₁ and B₂, are inputs which are respectively affiliated to different fuzzy membership functions. C₁ and C₂ are outputs respectively affiliated to different fuzzy membership functions. IF-THEN rules can be logically represented by fuzzy operators “and” (defined as MIN) and “or” (defined as MAX). Defuzzification is the opposite operation of fuzzification. It aims at calculating crisp value from a fuzzy output. There are some popular defuzzification techniques like middle of maximum, mean of maximum, center of area. Among them, calculating the centroid of area (COA) is the most popular [29]. Its formulation is shown in Equation (6), where μ (x) is the membership function of the aggregated output x. In the present study, membership functions are assumed to be triangular type considering it is simple.

$Z = \frac{\int_{x}^{x} μ (x) d x}{\int_{x}^{μ} (x) d x}$ (6)

3 Case study

3.1 Data collection

Data was collected from the mining site on the section from Km 4 + 400 to Km 5 + 250 of the project Motorway on Corridor Vc Section Počitelj-Zvirovići (Lot1) situated in Capljina, Bosnia and Herzegovina. The mentioned section is filled up by hills with granite as the principal rock. So, blasting is the most cost-effective way to excavation for this section.

The used dynamite was ammonium nitrate fuel oil (ANFO); the depth of holes ranges from 5 m to 13 m; the diameter of holes ranges from 72 mm to 82 mm; the holes in same row are connected by electronic detonators with a delay of 25 ms, and rows are connected by electronic detonators with a delay of 67 ms. Blasting vibration monitor Instantel Minimate Plus Model was used to record vibration waves and analyze data. The adopted sampling frequency was 4096 S/s, the duration for recording was 50 seconds. A view of the blasting on site can be accessed from Fig. 3. The original data collected by the present study has 5 dimensions except measured PPV, including Q(kg), B(m), T(m), W(kg), D(m), where Q means the total quantity of explosives. Totally, 280 samples were collected. 260 of them were used to train the proposed method and remaining samples were for validation.

Fig. 3

A view of the blasting in Project Motorway on Corridor Vc Section Počitelj-Zvirovići (Lot1).

3.2 Data clustering

260 samples were inputted into the KMM program with Matlab R2015b as the carrier. In the present study, each of the above six parameters is separately clustered into 4 groups, including Very Low, Low, High, and Very High. The outputs are shown in Table 2.

Table 2
The value ranges of outputted clusters

Param-eter Very low Low High Very high

W (0, 350) (350, 400) (400, 470) (470, +∞)

D (0, 1550) (1550, 1860) (1860, 2760) (2760, +∞)

Q (0, 100) (100, 200) (200, 300) (3000, +∞)

B (0, 3) (3, 5) (5, 7) (7, +∞)

T (0, 2) (2, 3.6) (3.6, 4.4) (4.4 +∞)

PPV (0, 0.60) (0.60, 1.20) (1.20, 2.15) (2.15 +∞)

Param-eter	Very low	Low	High	Very high
W	(0, 350)	(350, 400)	(400, 470)	(470, +∞)
D	(0, 1550)	(1550, 1860)	(1860, 2760)	(2760, +∞)
Q	(0, 100)	(100, 200)	(200, 300)	(3000, +∞)
B	(0, 3)	(3, 5)	(5, 7)	(7, +∞)
T	(0, 2)	(2, 3.6)	(3.6, 4.4)	(4.4 +∞)
PPV	(0, 0.60)	(0.60, 1.20)	(1.20, 2.15)	(2.15 +∞)

3.3 Minimum Redundancy Maximum Relevance

MRMR requires that the value of each parameter must be categorized before initialing calculation. KMM has undertaken such categorization, as shown in Table 2. If the value of a data point is located within a cluster, the data point is replaced with the numerical name of the cluster. For example, if W = 1000 kg is clustered as “Very Low”, it should be replaced with number 1. If it locates within “Low” cluster, number 2 will be used and so on.

Many studies have concluded that W and D are two fundamental parameters for ground vibration [3 , 9]. Thus, in the present study, any candidate parameter set is defaulted to contain both W and D. In the case, according to permutation and combination, excluding the set of W and D, there are 7 candidate sets. Matlab R2015b was employed to calculate the values of R, Dr, and MID for each candidate set, as shown in Table 3. Evidently, the set of W, D, B has the highest score. As a result, W, D, B were selected as the parameters for PPV prediction in this paper.

Table 3
The MRMR scores for candidate feature sets

Parameters R Dr MID

W, D, Q 0.311 0.835 0.524

W, D, B 0.233 0.963 0.730

W, D, T 0.239 0.958 0.719

W, D, Q, B 0.193 0.891 0.698

W, D, Q, T 0.263 0.887 0.624

W, D, B, T 0.270 0.983 0.713

W, D, Q, B, T 0.275 0.922 0.647

Parameters	R	Dr	MID
W, D, Q	0.311	0.835	0.524
W, D, B	0.233	0.963	0.730
W, D, T	0.239	0.958	0.719
W, D, Q, B	0.193	0.891	0.698
W, D, Q, T	0.263	0.887	0.624
W, D, B, T	0.270	0.983	0.713
W, D, Q, B, T	0.275	0.922	0.647

3.4 Mamdani fuzzy inference model development

Section 2.3 stated that triangular was selected to be the form of membership functions in the resent study. With the help of fuzzy inference tool provided by Matlab R2015b and the results in Table 2, the membership functions of three inputs and one output were obtained, as shown in Figs. 4–7, respectively. Both W and B have a positive correlation with PPV, while it is negative for D. Additionally, compared with W or D, B has weaker influence on PPV. Thus, with the suggestions from experts and experienced engineers, the principle of determining inference rules was set as follows: 1) if W or D jumps to a better state, PPV updates itself into corresponding state; 2) if B jumps from Very Low to Low or from High to Very High, PPV keeps unchanged; 3) if W = VeryLow and D=Very High and B = Very Low, then PPV = Very Low. For example, if W = VeryLow and D = High and B = Very Low, then PPV = Low; if W = Low and D = Very High and B = Very Low, then PPV = Low; if W = Low and D = Very High and B = Low, then PPV = Low; if W = Low and D = Very High and B = High, then PPV = High. Thus, 64 interference rules were developed in total. Finally, with the method of centroid of area, PPV was predicted, as shown in Fig. 8.

Fig. 4

The graphic presentation of membership function of W.

Fig. 5

The graphic presentation of membership function of D.

Fig. 6

The graphic presentation of membership function of B.

Fig. 7

The graphic presentation of membership function of PPV.

Fig. 8

The graphic view of output viewer of MFIS.

4 Comparison study

In order to verify the validation of the proposed method, the Mamdani Fuzzy Inference without parameter selection and membership functions analysis were conducted in term of PPV prediction accuracy. In which the collected data in this paper were directly inputted into MFIS with triangular as the type of membership functions. Additionally, USBM equation and multiple linear regression analysis (MLRA) were also considered to participate in the comparison. MLRA is designed to predict the value of dependent variable with various independent variables as basis [30]. Its representative equation is shown in Equation (7), where Y′ denotes the predicted variable which is related with independent variables [X₁, X₂, ⋯, X_i], the parameters d and [a₁, a₂, ⋯, a_i] are coefficients in the form of constant. To obtain the values of parameters contained by USBM equation, it was also changed into the form of Equation (8) through ln operation. Software SPSS 22 was used to perform regression analysis. The specified mathematical expressions of MLRA and USBM are shown in Equations (9) and (10), respectively. $Y' = d + a_{1} X_{1} + a_{2} X_{2} + \dots \dots + a_{i} X_{i}$ (7) $ln PPV = ln k - b ln D + 0.5 b ln W$ (8) $PPV = 0.00005 W - 0.009 D + 1.01 b - 3.17$ (9) $PPV = 16.81 {(D / \sqrt{W})}^{- 1.707}$ (10)

20 samples were collected to for validation. The values of predicted PPV are listed in Table 4.

Table 4

The predicted PPV obtained by using different techniques

Number	Measured PPV (cm/s)	Predicted PPV (cm/s)
		USBM	MLRA	MFIS	The proposed method
1	1.30	1.76	1.03	1.42	1.36
2	1.01	1.22	1.15	0.92	0.93
3	1.73	1.26	1.49	1.51	1.69
4	1.27	1.62	1.38	1.36	1.25
5	0.60	1.23	0.32	0.72	0.65
6	2.18	1.58	2.32	2.14	2.23
7	1.52	1.49	1.84	1.56	1.49
8	2.03	1.34	1.96	1.84	1.96
9	0.70	1.46	0.93	0.74	0.82
10	1.91	2.24	2.15	2.05	1.94
11	2.09	2.41	2.26	2.14	2.02
12	2.88	3.01	2.56	2.80	2.93
13	2.97	2.14	3.22	3.08	3.01
14	2.91	3.29	2.81	2.74	2.82
15	1.18	1.12	1.26	1.34	1.21
16	1.57	1.95	1.36	1.67	1.48
17	1.62	2.24	1.89	1.54	1.65
18	1.84	1.31	1.62	1.81	1.71
19	1.58	1.78	1.39	1.68	1.63
20	1.59	1.81	1.44	1.71	1.54

Fig. 9 is the line chart for the values in Table 5. In Fig. 9, red line connects measured PPVs together, pink line is for the values of PPVs predicted by USBM, green line marks the predicted PPVs resulting from MLRA, black line strings the predictions made by MFIS, and the predicted PPVs from the proposed method are labeled in blue. It can be seen that blue line is consistent with the red line to a best degree.

Fig. 9

The line chart for the predicted PPVs resulting from different methods.

5 Evaluation of model performance

The evaluation here aims at checking the prediction performance of the proposed method. Three indicators were employed to make such evaluation, including coefficient of correlation (R²) [30], root-mean-square error (RMSE) [30], mean absolute error (MAE) [30]. The mathematical expressions of the mentioned indicators are shown in Equation (11), Equation (12), and Equation (13), respectively, where x_i and x′_i mean the value of the i^th measured and predicted PPV, respectively, n represents the total number of samples, x_mean denotes the mean of values of measured PPVs. If a method is excellent, its R², RMSE, and MAE should approach to 1, 0, and 0, respectively [29, 30]. $R^{2} = \frac{\sum_{i = 1}^{n} {(x_{i} - x_{mean})}^{2} - \sum_{i = 1}^{n} {(x_{i} - x'_{i})}^{2}}{\sum_{i = 1}^{n} {(x_{i} - x_{mean})}^{2}}$ (11) $RMSE = \sqrt{\frac{\sum_{i = 1}^{n} {(x_{i} - x'_{i})}^{2}}{n}}$ (12) $MAE = \frac{\sum_{i = 1}^{n} | x_{i} - x'_{i} |}{n}$ (13)

Each of the four methods compared in section 4 was evaluated in term of the three indictors, as shown in Table 5. Evidently, the proposed method has better values, i.e. R² = 0.989, RMS = 0.264, VAF = 0.059. Additionally, under the given values of R², the graphic comparison between measured PPV and predicted PPV by USBM, MLRA, MFIS, and the proposed method is listed in Figs. 10–12, and 13, respectively. It confirms that the proposed method has excellent performance in predicting PPV induced by blasting.

Table 5

The values of R², RMSE, and MAE of the four methods compared

Method	R ²	RMSE	MAE
USBM	0.479	1.834	0.410
MLRA	0.891	0.894	0.200
MFIS	0.968	0.467	0.105
The proposed method	0.989	0.264	0.059

Fig. 10

The comparison of measured and predicted PPV by USBM equation.

Fig. 11

The comparison of measured and predicted PPV by MLRA.

Fig. 12

The comparison of measured and predicted PPV by MFIS.

Fig. 13

The comparison of measured and predicted PPV by the proposed method.

6 Conclusion

A method based on Mamdani Fuzzy Inference System was developed by the present study to predict the PPV induced by blasting. Minimum Redundancy Maximum Relevance was employed to select the blasting design parameters which significantly contribute to PPV. K-means method was applied to determine the value ranges of selected parameters. Then, the selected variables and value ranges are connected to be the membership functions of MFIS. Totally, 280 samples were collected from a blasting construction site. Parameter selection identified three effective variables including maximum charge used per delay (W), burden (B), and the distance from the monitoring point to the blasting face (D). The present study compared the proposed method, empirical equation USBM, MLRA, and MFIS in terms of the difference between predicted PPV and measured PPV, coefficient of correlation (R²), root-mean-square error (RMSE), and mean absolute error (MAE). The results from that verified the validation of the proposed method.

Conflict of interest

The authors declare that they have no conflict of interest.

Footnotes

Acknowledgments

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

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