Introduction

In rock quarry blasting, only 20–30 % of the energy produced by explosives is utilized to fragment and displace the rock mass. The rest of the energy is wasted and produces undesirable environmental impacts such as ground vibration, air overpressure (AOp), flyrock, and back-break (Segarra et al. 2010; Monjezi et al. 2012; Raina et al. 2014; Jahed Armaghani et al. 2014; Marto et al. 2014; Ebrahimi et al. 2015). Various empirical predictors have been established to predict ground vibration and AOp induced by blasting. However, such approaches only consider limited numbers of parameters influencing ground vibration and AOp, although these phenomena are also affected by other controllable or uncontrollable parameters such as blast geometry and geological conditions (Douglas 1989; Singh and Singh 2005). As a result, in many cases, empirical methods are not accurate enough, while prediction of the ground vibration and AOp with a high degree of accuracy is important to estimate the blasting safety area.

In addition to the empirical equations, the use of statistical methods such as simple and multiple regression techniques for ground vibration and AOp prediction has received attention mainly due to their ease of use (Hudaverdi 2012). Apart from the statistical methods, the use of soft computing techniques for prediction of ground vibration and AOp recently has been highlighted in the literature (Khandelwal and Singh 2006; Monjezi et al. 2010, 2011; Mohamed 2011).

Artificial neural networks (ANNs), known as flexible nonlinear function approximations, have been widely used to analyze engineering problems. Although ANNs are able to directly map input to output patterns and utilize all influential parameters in the prediction of ground vibration and AOp, there are still some limitations: the slow rate of learning and the possibility of becoming trapped in local minima (Eberhart and Shi 1998; Eberhart et al. 1996; Tsou and MacNish 2003; Hajihassani et al. 2014). To address these limitations, the use of powerful optimization algorithms such as particle swarm optimization (PSO) is advantageous for improving ANN performance (Kennedy and Eberhart 1995; Adhikari and Agrawal 2011; Jahed Armaghani et al. 2014; Momeni et al. 2015). This study presents a novel approach founded on a hybrid PSO-based ANN model to predict ground vibration and AOp induced by blasting.

Blast vibrations

Blasting operations may cause excessive environmental impacts such as ground vibration, AOp, dust, and flyrock (Bhandari 1997). High blast vibrations can cause damage to structures and nearby residential areas (Konya and Walter 1990; Singh and Singh 2005). In the following parts, effective parameters and relevant previous investigations of these vibrations are discussed comprehensively.

Ground vibration

Ground vibration is a wave motion which travels away from the blast to nearby areas (Khandelwal and Singh 2009). A considerable amount of energy is applied in every ground vibration in which this energy is supposed to be applied to rock fracturing. The problems caused by ground vibration include large impacts on the structures, groundwater, and ecology of the nearby area (Khandelwal and Singh 2009; Duvall et al. 1963; Ghasemi et al. 2013).

When an explosive is detonated in a blasthole, the chemical reaction of the explosives produces a high-pressure and high-temperature gas. This gas pressure crushes the rock adjacent to the blasthole. The detonation pressure decays or dissipates quickly. A wave motion is created in the ground by the strain waves conveyed to the surrounding rocks (Duvall and Petkof 1959). Due to various breakage mechanisms like radial cracking, crushing, and reflection breakage in the free face, the strain energy carried by these strain waves fragments the rock mass. During and after the stress wave propagation, high-pressure high-temperature gases extend radial cracks and any discontinuity, fracture, or joint (Dowding 1985). The strain waves propagate as elastic waves when the stress wave intensity diminishes to the level where no permanent deformation occurs in the rock mass (see Fig. 1). These waves are identified as ground vibration. The ground vibration spreads from the blasthole in all directions (Dowding 1985) and can cause damage to buildings and other structures (Siskind et al. 1980).

Fig. 1
figure 1

Ground vibration due to blasting (Bhandari 1997)

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Many parameters are involved in ground vibration such as the blasting design, the distance between the free face and the monitoring point, and geological conditions (Wiss and Linehan 1978; Khandelwal and Singh 2006; Ghoraba et al. 2015). It is essential to optimize the blasting design parameters to decrease ground vibration based on the properties of the rock mass, which include rock strength, density, wave velocity, and discontinuity conditions (Singh and Sastry 1986). The ground vibration can be measured in terms of the peak particle velocity (PPV) and frequency. In the Indian Standard Institute (Bureau of Indian Standard 1973) and German DIN Standard 4150 (New 1986), the PPV is considered as a vibration index, which is a significant indicator for controlling structural damage. PPV due to ground vibration in surface blasting is a significant parameter for the prediction of ground vibration. PPV principally depends on two parameters: the maximum charge used per delay and the distance from the free face (Ozer et al. 2011; Basu and Sen 2005).

Soft computing techniques have been widely used by several researchers to predict PPV. Singh and Singh (2005) employed ANN and regression analysis to predict PPV. In their study, the hole diameter, number of holes, hole depth, burden, spacing, and the distances from the free face were considered as input parameters. They demonstrated that ANN is a more accurate approach compared to regression analysis for predicting ground vibration. Khandelwal and Singh (2006) investigated four widely used empirical predictors to estimate the PPV for 150 blast datasets and compared the computed results with actual field data. Subsequently, they developed an ANN with two inputs (maximum charge per delay and distance from free face) and one output (PPV). They found that ANN results are more accurate compared to empirical predictors. Iphar et al. (2008) utilized two different methods including simple regression and the adaptive neuro-fuzzy inference system (ANFIS) to predict PPV induced by blasting. They used 44 PPV values obtained from blasting operations in Turkey. Their results showed that the ANFIS model yielded better results in comparison to regression analysis. Khandelwal and Singh (2009) used ANN and multivariate regression analysis (MVRA) techniques to predict PPV and frequency by incorporating rock properties, blast design, and explosive parameters. A total of 174 vibration records were used to predict PPV and frequency with ten input parameters. The ANN results indicated closer agreement with the field datasets as compared to MVRA prediction. An ANN model with four input parameters—maximum charge per delay, distance from the free face to the monitoring point, stemming length, and hole depth—was developed by Monjezi et al. (2011) to predict PPV. A database consisting of 182 datasets was collected at different strategic and vulnerable locations around the Kandovan tunnel in Iran. They demonstrated that ANNs are applicable tools to predict blast-induced ground vibration. In addition, from the sensitivity analysis they found that the distance from the free face has the most influence on PPV, while stemming has the least. Fişne et al. (2011) utilized a fuzzy logic approach and classical regression analysis to predict PPV using 33 datasets obtained from the Akdaglar quarry in Turkey. In their research, the charge weight and distance from the free face were considered as input parameters to predict PPV. They concluded that the predicted PPVs obtained from the fuzzy model were much closer to the measured values in comparison to those predicted by the statistical model. Monjezi et al. (2013) predicted PPV using different empirical equations and the ANN technique. They compared the computed results with the actual field data obtained from Shur River Dam in Iran. Total charge, maximum charge per delay, and distance between the shot point and the monitoring station were considered as input parameters for the prediction of PPV. They found that the ANN model was more accurate in comparison to the empirical equations. In another study on PPV prediction, Hajihassani et al. (2014) used a combination of an imperialism competitive algorithm (ICA)-ANN to predict PPV values obtained from Harapan Ramai granite quarry. In their study, the burden-to-spacing ratio, stemming length, maximum charge per delay, Young’s modulus, p-wave velocity, and distance from the free face were utilized as model inputs. They concluded that the ICA-ANN approach can predict PPV with higher accuracy compared to empirical equations.

Air overpressure

The explosion (blast) is produced by the shock wave of a chemical reaction where the pressure of reactive gases reaches sonic velocity (Baker et al. 1983). The gas pressure velocity increases rapidly in the blasthole. Consequently, the pressure in the blasthole suddenly loads the surrounding rocks, which move away from the borehole. The pressure in terms of blasting is mainly considered using shock and gas mechanisms (Roy 2005).

AOp is created by a large wave from the explosion point to the free surface. Hence, the AOp is a wave which is refracted horizontally by density variations in the atmosphere. AOp atmospheric pressure waves comprise an audible high-frequency and a sub-audible low-frequency sound. Generally, four important sources can cause AOp waves in blasting operations: the air pressure pulse, which results from displacement of the rock at the bench face as the blast progresses; the rock pressure pulse, which is induced by ground vibration; the gas release pulse, which results from the escape of gases through rock fractures; and the stemming release pulse, which results from the escape of gases from the blasthole when the stemming is ejected (Siskind et al. 1980; Wiss and Linehan 1978; Morhard 1987).

According to Kuzu et al. (2009), AOp is identified in terms of sound and measured in decibels (dB) or pascals (Pa), where 20 Hz is the lowest sound detectable by the human ear. Hence, it is undeniable that there is a possibility of concussion in the human ear with sound at more than 20 Hz. In addition, structural damage may occur at an AOp level of 180 dB, general window breakage occurs at 171 dB, and occasional window breakage occurs at 151  dB (Kuzu et al. 2009). According to Siskind et al. (1980), as reported by the United States Bureau of Mines (USBM), a value of 134 dB is recommended for AOp limitation. Therefore, many attempts have been made to control AOp values (Kuzu et al. 2009; Rodriguez et al. 2010).

Several parameters affect AOp in blasting operations. According to Khandelwal and Kankar (2011), blast geometry, explosive charge weight per delay, distance between the free face and the monitoring point, geological discontinuities, blasting direction, surface topography, and vegetation are the foremost parameters influencing AOp. Konya and Walter (1990) found that AOp can be controlled by the type and length of stemming materials. Their findings reveal that a stemming particle size of about 0.05 times the blasthole diameter provides the best confinement and the materials have to be angular to function properly. Moreover, other parameters affect AOp, such as overcharging, weak strata, atmospheric conditions, and secondary blasting (Siskind et al. 1980; Griffiths et al. 1978; Dowding 2000). However, AOp induced by blasting is not easy to predict as the same blast design can produce different results in different cases.

Based on the parameters that influence AOp, many attempts have been made to establish correlations for the prediction of AOp induced by blasting. Rodríguez et al. (2007) developed a semi-empirical model for the prediction of the air wave pressure outside a tunnel due to blasting. This method was tested in several cases and it was proven that it can be used under different conditions. Kuzu et al. (2009) established a new empirical relationship between AOp and two other parameters (distance between free face and monitoring point and maximum charge per delay), which are the most important variables for AOp. They used 98 AOp records from quarry blasting operations under different conditions and demonstrated that the proposed equation predicted AOp with reasonable accuracy. Segarra et al. (2010) provided a new AOp prediction equation based on monitoring data obtained from two quarries. Blasting data and AOp measurements were obtained from 122 records of 40 blasting operations in rocks with low to very low strength. They concluded that the accuracy of AOp prediction was 32 %. In addition, the proposed model was validated using five new blasting data with 22.6 % accuracy.

Apart from empirical equations, some soft computing methods have been developed to predict AOp. Khandelwal and Singh (2005) presented an ANN model to predict AOp from two variables including distance (between free face and monitoring point) and sound pressure level. They compared the ANN results with the USBM predictor and multivariate regression analysis (MVRA) results. The comparison showed that ANN yielded better estimates compared to USBM and MVRA predictors. Mohamed (2011) predicted the AOp using a fuzzy inference system and ANN using two parameters: maximum charge per delay and distance from the free face to the monitoring point. He compared the results with the values obtained by regression analysis and observed field data, and concluded that the ANN and fuzzy models gave more accurate prediction compared to regression analysis. Khandelwal and Kankar (2011) predicted AOp due to blasting using 75 datasets obtained from three mines by the support vector machine (SVM) method. They compared the AOp values predicted by SVM with the results of a generalized predictor equation. Using the maximum charge per delay and distance from the free face to the monitoring station as input parameters, they showed that the values of AOp predicted by SVM were much closer to the actual values as compared to the values predicted by the generalized predictor equation. Tonnizam Mohamad et al. (2012) used ANN to predict AOp using 38 datasets obtained from blasting operations. In their study, the hole diameter, hole depth, spacing, burden, stemming, powder factor, and number of rows were used as input parameters. Their results show the applicability of the proposed model to the prediction of AOp.

PPV and AOp prediction methods

Estimating a safe zone for blasting operations is an important subject in the field of geotechnical engineering, and prediction of blasting environmental impacts such as PPV and AOp before blasting operations is always necessary. Many attempts have been made to predict PPV and AOp using empirical methods. In the following sections, a brief review of these empirical methods is presented.

PPV prediction methods

Many researchers established empirical vibration equations to predict PPV (Duvall and Petkof 1959; Bureau of Indian Standard 1973; Langefors and Kihlstrom 1963; Davies et al. 1964; Ghosh and Daemen 1983; Roy 1993). In most of these equations, the maximum charge per delay and distance from the free face are considered as the main influential parameters for PPV prediction. It is well known that PPV is influenced by other factors such as blast geometry, rock strength, and discontinuity conditions which have not been incorporated explicitly in any of the empirical equations. So, different equations give different PPV values for the same blasting operation and there is no uniformity among the results predicted by different equations. Table 1 illustrates the PPV equations proposed by different researchers.

Table 1 Empirical PPV predictor presented by different researchers
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AOp prediction methods

Some empirical equations have been suggested to predict AOp. According to the National Association of Australian State Road Authorities 1983, AOp from confined blasthole charges can be estimated from the following empirical formula:

$$P = \frac{{140 \sqrt[3]{\frac{E}{200}}}}{d},$$
(1)

where P is the AOp (kPa), E the mass of charge per delay (kg), and d the distance from the free face (m). McKenzine (1990) recommended an equation to describe the decay of AOp as follows:

$${\text{dB}} = 165 - 24 { \log }(D/W^{1/3} ),$$
(2)

where dB is the decibel reading with a linear of flat weighting, D the distance between the free face and the monitoring point (m), and W the explosive charge weight per delay (kg).

Applying the cube root scaled distance factor (SD), in the absence of monitoring equipment, is another method of estimating the blast-induced AOp. The correlation between explosive charge weight per delay, distance, and SD is given as follows:

$${\text{SD}} = DW^{ - 0.33} ,$$
(3)

where D denotes the distance (m or ft), W the explosive charge weight (kg or lb), and SD the scaled distance (m kg−0.33 or ft lb−0.33).

Through the availability of sufficient data, establishment of the relationship between the values of SD and AOp is possible. A site-specific AOp attenuation formula can be developed when statistical analysis techniques can practically represent AOp data (White and Farnfield 1993; Rosenthal and Morlock 1987; Cengiz 2008). The prediction equation is shown as follows:

$${\text{AOp}} = H({\text{SD}})^{ - \beta } ,$$
(4)

where AOp is measured in pascals or decibels, H and β are the site factors, and SD is the scaled distance factor as given in Eq. (3). The scaled distance factor is widely used in surface blasting to predict AOp (Kuzu et al. 2009; Hustrulid 1999). The site factor values, H and β, for different blasting conditions are tabulated in Table 2.

Table 2 Site factors H and β for different blasting conditions
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Hybrid PSO-based ANN

Many researches have been conducted to improve the performance and generalization capabilities of ANNs. Ordinary ANNs employ the backpropagation (BP) algorithm in the learning process, which is a local search learning algorithm; therefore, the learning process of ANNs might cause the convergence of the solution to fail (Liou et al. 2009). Since PSO is a robust global search algorithm, it can be used to adjust the weights and biases of an ANN to increase the performance and accuracy. The following sections describe the procedure of ANN and PSO in minimization problems and the implementation of hybrid PSO-based ANN models.

Artificial neural network

An artificial neural network (ANN) can be identified as a simplified mathematical model of reasoning based on the human brain. ANN is able to determine the complex relationship among variables for the simulation of one (or more) output(s) (Specht 1991). A specific ANN model can be defined using three important components: the transfer function, network architecture, and learning rule (Simpson 1990). Based on the type of problem, these components need to be defined as an initial set of weights and display how weights should be modified during training to increase the performance (Monjezi and Dehghani 2008). The multilayer perceptron (MLP) is one of the most well-known feedforward neural network models and typically contains an input layer of source neurons, at least one hidden layer of computational neurons, and one output layer. Each of these layers has its own specific function. The input layer accepts inputs from the outside world and distributes them to the subsequent layers. Features hidden in the input patterns are detected by the neurons in the hidden layer. The output layer exploits these features to determine the output pattern (Bounds et al. 1998).

Several algorithms have been recommended for the training of neural networks. The BP is the most popular learning method among a vast number of MLP learning algorithms (Basheer and Hajmeer 2000). In the BP method, the input data are presented to the input layer to be propagated through the network until an output is generated. Each neuron determines its net weighted input using the following equation:

$$X = \sum\limits_{i = 1}^{n} {\mathop x\nolimits_{i} } \mathop w\nolimits_{i} - \theta ,$$
(5)

where n is the number of inputs, and x i and w i denote the values of the ith input and weight, respectively. The threshold applied to the neurons is denoted by θ. This input value passes through one of the activation functions such as a sigmoid, step, or linear function. Such a procedure is technically known as a learning or training procedure. The network computes its actual outputs, its weights, and a mathematical function model threshold. Afterward, the actual output is compared to the historical outputs to calculate the output error (Rafiai and Jafari 2011). The obtained error is propagated back through the network and updates the individual weights. This process is called the backward pass. This procedure is repeated until the error reaches a defined level such as the mean square error (MSE) (Simpson 1990; Kosko 1994). However, for training an ANN model, an experimental database requires an appropriate number of datasets (Dreyfus 2005).

Particle swarm optimization

The particle swarm optimization (PSO) algorithm originated from the social behavior of organisms (individuals) in swarms like flocks of birds (Kennedy and Eberhart 1995). PSO is an evolutionary population-based optimization technique that can be used to solve global optimization problems within a nonlinear procedure. In PSO algorithm, each particle denotes candidates’ solution to the optimization problem. In this algorithm, particles flow throughout the multidimensional search space to find the best solution. Therefore, in each optimization problem, several particles should be produced and scattered in the search space. The particles change their positions in the search space based on their experiences and those of neighboring particles, and therefore the particles make use of their own experience and those of their neighbors (Engelbrecht 2007). These particles form a population which is technically known as a swarm.

Finding the best solution using PSO starts with initialization of random particles (solutions) which are assigned random positions and velocities. Subsequently, the algorithm searches for the best solution through an iterative procedure (Eberhart and Shi 2001). In the process each particle keeps track of its best position, known as its personal best (p best), as well as the overall best value accomplished by other particles in the swarm, known as the global best (g best).

Through the learning process, a particle’s journey toward both the p best and the g best position is speeded up by calculating a new velocity. The new velocity is calculated in terms of the particle’s distance from the p best and g best positions, which will affect the particle’s next position in the next iteration.

A relatively simple procedure is required to obtain the optimized solution using PSO as compared to the other optimization algorithms (Van den Bergh and Engelbrecht 2000). In fact, PSO operates based on two simple equations (Eqs. 6 and 7) for updating the particles’ velocities and positions. To increase the convergence rate of the algorithm, an inertia weight can be used in the original equations (Shi and Eberhart 1998), as in Eq. (6). The inertia weight determines the rate of contribution of a particle’s previous velocity to its current velocity:

$$\overrightarrow {{v_{\text{new}} }} = w \cdot \vec{v} + r_{ 1} C_{1} \times \left( {\overrightarrow {{p_{\text{best}} }} - \vec{p}} \right) + r_{ 2} C_{2} \times \left( {\overrightarrow {{g_{\text{best}} }} - \vec{p}} \right),$$
(6)
$$\overrightarrow {{p_{\text{new}} }} = \vec{p} + \overrightarrow {{v_{\text{new}} }}$$
(7)

where \(\overrightarrow {{v_{\text{new}} }}\) is the new velocity and w is the inertia weight. \(\vec{v}\), \(\overrightarrow {{p_{\text{new}} }}\), and \(\vec{p}\) are the current velocity, new position, and current position of particles, respectively, \(C_{1 }\) and \(C_{2}\) are acceleration constants, \(\overrightarrow {{ p_{\text{best}} }}\) is the personal best position of the particle, \(\overrightarrow {{g_{\text{best}} }}\) is the globally best position among all particles, and \(r_{ 1}\) and r 2 are random values in the range (0, 1) sampled from a uniform distribution.

PSO-based ANN algorithm

Many attempts have been made to improve the ANN performance by means of optimization algorithms, due to the fact that an optimum search process of conventional ANNs might fail and return an unsatisfactory solution (Liou et al. 2009; Engelbrecht 2007). Several studies have been conducted to investigate the ability of PSO as a training algorithm for a number of different ANN architectures. Eberhart and Kennedy (1995) presented the first results of utilizing the basic PSO to train ANNs, whereas several scholars have further demonstrated the capability of PSO in training ANNs and showed that the PSO is an effective alternative for training ANNs (Mendes et al. 2002; Settles and Rylander 2002; Gudise and Venayagamoorthy 2003). It should be noted that further attempts have been made to employ other optimization techniques in training ANNs, for example, genetic algorithm and ant colony optimization techniques (Montana and Davis 1989; Socha and Blum 2007). Nevertheless, it has been proven that ANNs trained by PSO provide more accurate results compared to other learning algorithms (Engelbrecht 2007).

In ANN training, a set of weights and biases are determined which minimize an objective function such as MSE. So, MSE can be used as the fitness function in training an ANN using PSO, due to the fact that a fitness function is required to generate a PSO-based ANN model.

In a minimization problem, there is one global minimum and a number of local minima. ANN searches for a solution in the local region due to its inherent property and therefore usually gets trapped in a local minimum. PSO has a competent capability to search the entire search space to find the global minimum and continues searching around it. Hence, a hybrid PSO-based ANN model has the search properties of both PSO and ANN, where PSO looks for the global minimum in the search space and ANN uses the global minimum to find the best results (Gordan et al. 2015).

The learning process in a PSO-based ANN model is initialized by generating a group of random particles in which each particle represents a set of weights and biases in the model. The PSO-based ANN model is trained using the initial weights and biases (i.e., initial position of particles), the particle’s velocity and position are updated using the PSO equations, and subsequently, in each iteration, the weights and biases of the model are adjusted. In each iteration, the MSEs between the actual and predicted values are calculated and the errors are reduced by changing the positions of the particles. This process is continued to find the best weights and biases for an ANN to minimize the error function.

Case study and data collection

This study was conducted at Hulu Langat quarry site in Selangor State, Malaysia. Geographically, the quarry lies at a latitude of 3°7′0″N and a longitude of 101°49′1″E and is located in the south of Selangor. An overall view of the Hulu Langat quarry site is shown in Fig. 2. This quarry is composed of granitic rocks with the capacity to produce large amounts (between 280,000 and 360,000 tons per month) of aggregate. Blasting is carried out 10–12 times per month, depending on the weather conditions. All blasting operations are conducted using blastholes 89 mm in diameter. Ammonium nitrate and fuel oil (ANFO) and dynamite were used as the main explosive material and for initiation, respectively. The blastholes were stemmed using fine gravels.

Fig. 2
figure 2

A view of the Hulu Langat granite quarry site

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During 9 months from August 2012 to April 2013, 88 datasets were collected. During data collection, blasting parameters including hole depth, maximum charge per delay, burden, spacing, stemming length, sub-drilling, powder factor, and number of holes were obtained. In each blast, PPV and AOp values were recorded using a VibraZEB seismograph. In the case of AOp, the values were monitored during each blasting operation using linear L type microphones connected to the AOp channels. A range of AOp values from 88 dB (7.25 × 10−5 psi or 0.5 Pa) to 148 dB (0.0725 psi or 500 Pa) can be recorded by VibraZEB. The microphones have an operating frequency response from 2 to 250 Hz, which is adequate for measuring AOp accurately in the frequency range critical for structures and human hearing. All AOp and PPV values were recorded in front of the quarry bench and approximately perpendicular to it.

The crushing plant and workshops are located about 400 m to the southwest of the quarry face, while the nearest residential area is about 800 m to the west of the quarry face. Therefore, the distance between the monitoring point and free face was set as 300, 600, and 700 m. Figure 3 shows the location of the quarry site and nearest residential area.

Fig. 3
figure 3

Location of the quarry site and the nearest residential areas

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The use of the SD is a common technique to predict PPV and AOp values resulting from blasting. The relationships between the SD and the two parameters of distance and explosive charge weight per delay are formulated as follows for PPV prediction (Duvall and Petkof 1959):

$${\text{SD}} = DW^{ - 0.5} ,$$
(8)

where W is the maximum charge per delay (kg) and D represents the distance between the monitoring point and free face (m). The correlation of maximum charge weight per delay, distance, and SD is given in Eq. (3) for AOp prediction. Afterward, PPV and AOp values can be determined using the USBM-suggested equation as follows:

$${\text{PPV}}/{\text{AOp}} = K ({\text{SD}})^{B} ,$$
(9)

where B and K are site constants. The graphs of the measured PPV and AOp values against their SDs are shown in Figs. 4 and 5, respectively. In addition, two empirical equations were proposed for the prediction of PPV and AOp values as indicated in these figures. Coefficient of determination R 2 values equal to 0.581 and 0.410 for PPV and AOp prediction suggest that the proposed equations can predict them with good accuracy level.

Fig. 4
figure 4

Relationship between scaled distance and PPV values

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Fig. 5
figure 5

Relationship between scaled distance and AOp values

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Development of PSO-based ANN model for PPV and AOp prediction

A MatLab code was developed to predict PPV and AOp using a hybrid PSO-based ANN model. ANNs work based on given data and do not have previous knowledge about the subject of prediction. Therefore, to predict the PPV and AOp induced by blasting, all relevant parameters should be determined. The following sections describe the development procedure of a PSO-based ANN model to predict the PPV and AOp induced by blasting.

Input and output parameters

Determining the input parameters is the first step of developing a prediction model for PPV and AOp. To develop a comprehensive and accurate model, the parameters with the greatest influence on PPV and AOp should be determined. In determining the influential parameters, it should be considered that the selected parameters must represent the site conditions as well as the blast design parameters, must be measurable, and must be easy to obtain concurrently.

A direct relationship exists between the blast design parameters and the PPV and AOp values in blasting operations. Therefore, the blast design parameters including hole depth, maximum charge per delay, burden-to-spacing ratio, stemming length, sub-drilling, powder factor, and number of holes were taken into account in modeling. The values of PPV and AOp may increase if the design of these parameters is carried out improperly.

Geological discontinuities, in addition to the aforementioned parameters, have a significant impact on PPV and AOp in blasting operations. In the presence of geological discontinuities, explosive gases escape intensely from the discontinuities, leading to high vibration magnitudes. Therefore, as a degree of jointing or fracturing in a rock mass, the rock-quality designation (RQD) was used in modeling to represent the geological discontinuities.

It is obvious that the PPV and AOp values decrease as the distance between the free face and the monitoring point increases. Therefore, as an influential parameter, this parameter was used in modeling. Table 3 shows the input and output parameters and their ranges. The modeling procedure was started by normalization of the input and output data. It is recommended that the input and output data be normalized before they are presented to the network. According to Rafig et al. (2001), normalization helps to improve the learning speed of the network. By using the following equation, the data were normalized into a range of −1 to 1.

Table 3 Input and output parameters used in the prediction model
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$$xN = \left( {\frac{{x - {\text{Min}} \;x}}{{{\text{Max}} \;x - {\text{Min}} \;x}}} \right){ \times }2 - 1,$$
(10)

where xN is the normalized value of the variable x, and Min x and Max x are the minimum and maximum values, respectively, for the variable x.

Network design

A PSO-based ANN model performs best when its parameters are selected properly. The PSO parameters (including number of particles, acceleration constants for gbest (C 1) and pbest (C 2), and inertia weight) as well as the ANN parameters (network architecture including the number of hidden layers and the number of nodes in a hidden layer) are related to the performance of the PSO-based ANN model. Therefore, many computations were conducted to determine the optimal configuration of the PSO-based ANN model. A series of sensitivity analyses was conducted to find the optimum PSO parameters. Subsequently, the optimum network architecture was determined using the trial and error method as well as the K-fold cross-validation technique.

PSO parameters

A MatLab code was developed to perform the sensitivity analyses. These analyses consist of several independent steps to determine the optimum number of particles, acceleration constants, and inertia weight. The performance of PSO-based ANN models in minimizing the MSE was evaluated during the sensitivity analyses.

As an initial model, a PSO-based ANN model consisting of a single hidden layer with nine nodes was used. For each analysis, 80 % of the data was assigned for training while the remaining 20 % was used for testing. Each analysis was conducted three times and the best value was selected as the representative value of the model.

To obtain the appropriate number of particles in the swarm (swarm size), a series of sensitivity analyses was applied to the PSO swarm size because no other method of finding the optimum swarm size exists. While a small swarm may fail to converge to a global solution, a large swarm may lead to delay in the convergence and decrease the efficiency. The analyses were performed by setting a fixed iteration number of 1000 for each model with various numbers of particles and a fixed value of 2 for both acceleration constants, C 1 and C 2. R 2 and MSE are the model selection criteria. Figure 6 illustrates the results of the sensitivity analyses for the number of particles.

Fig. 6
figure 6

a R 2 for models with different swarm size, b MSE for models with different swarm sizes

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According to Fig. 6a, in general, the values of R 2 have been increased by increasing the number of particles. However, after a significant increase in the values of R 2 when the number of particles increased from 10 to 250, the network performance does not improve afterward. The same results were obtained between the number of particles and the values of MSE, as shown in Fig. 6b. Meanwhile, the training time was gradually increased by increasing the number of particles, as shown in Fig. 7. As a result, a swarm size of 250 was selected as the optimum number of particles to avoid ineffectual iterations in the models.

Fig. 7
figure 7

Training consumed time for models with different swarm sizes

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The next sensitivity analyses were conducted to determine the optimum values of the acceleration constants, C 1 and C 2. A series of candidate combinations were used based on the original (Kennedy and Eberhart 1995) and modified (Clerc and Kennedy 2002) acceleration constants, as shown in Table 4. The analyses were conducted using the obtained optimum swarm size of 250 on the initial network, including a single hidden layer with nine nodes. The results of the sensitivity analyses are shown in Table 4. Superior results were obtained by model number 3 compared to other models, as it had the highest values of R 2 and the lowest values of MSE for the training and testing datasets. As a result, the values of 1.714 and 2.286 were selected for acceleration constants C 1 and C 2 to be used in the prediction of PPV and AOp.

Table 4 The results of sensitivity analyses for acceleration constants C1 and C2
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The next step in designing an optimum network is finding an appropriate inertia weight. Based on the inertia weight suggested in previous studies (Shi and Eberhart 1998; Clerc and Kennedy 2002), four tests with different inertia weights were designed to find the optimum value of the inertia weight in the PSO equation. The same initial swarm with a size of 250 was applied in all the tests, and the acceleration constants were set at the previously defined optimum values of 1.714 and 2.286 for C 1 and C 2, respectively. Figures 8 and 9 show the values of R 2 and MSE for the training and testing datasets at different inertia weights. According to these figures, the highest R 2 and the lowest MSE for the training and testing datasets were obtained at an inertia weight of 0.5. Hence, this value was selected as the optimal inertia weight.

Fig. 8
figure 8

R 2 for training and testing datasets at different inertia weights

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Fig. 9
figure 9

MSE for training and testing datasets at different inertia weights

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Network architecture

PSO can only adjust the weights and biases of a model to minimize the learning error. Therefore, the network architecture, composed of the number of hidden layer(s) and the number of nodes in each hidden layer, should be determined through the trial and error method as a consequence of the fact that there is no absolute method of determining the optimum network architecture.

Following the determination of the PSO parameters, the optimal network architecture was obtained. This was done through the trial and error method. A K-fold cross-validation technique (Diamantidis et al. 2000) was employed to evaluate the performance of each model. In this technique, the data are divided into K parts, of which K − 1 parts are used for training and one part is used to test the model. The process is repeated K times and therefore all of the data are used in the training and testing steps.

To determine the optimal network architecture, 14 hybrid models were considered and a fivefold cross-validation was used to evaluate the performance of the models. Each model was trained with fourfold cross-validation (70 datasets) and tested with the rest of the data (18 datasets). Therefore, each model was trained and tested five times with different combinations of training and testing datasets. The values of R 2ave and MSEave for the testing datasets were considered as the model performance criteria.

The processes were conducted with different numbers of hidden layers in the networks and various numbers of nodes in each hidden layer. Hidden layers of one and two layers were considered to find the optimum number of hidden layer(s) and 6, 9, 12, 15, 18, 21, and 24 nodes were considered to find the optimum number of nodes in each hidden layer. All models were trained with the optimized PSO parameters obtained in previous analyses. The results of the analyses are tabulated in Table 5.

Table 5 Performance of trained PSO-based ANN models
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Figures 10 and 11 display the values of R 2ave and MSEave for trained models with different architectures. According to the figures, model number 3 presents the best performance in terms of values of R 2ave and MSEave for the testing datasets among all models: 0.89 for R 2ave and 0.038 for MSEave. Therefore, the architecture of model number 3 was selected as the optimum architecture. The structure of the selected PSO-based ANN model consisting of one hidden layer and 12 nodes in the hidden layer is illustrated in Fig. 12.

Fig. 10
figure 10

R 2ave for trained PSO-based ANN models

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Fig. 11
figure 11

MSEave for trained PSO-based ANN models

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Fig. 12
figure 12

Structure of the selected PSO-based ANN model to predict PPV and AOp induced by blasting

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Analysis of the results

A graphical comparison between the measured and predicted values of PPV and AOp employing different training datasets is shown in Fig. 13. According to the figure, a superior concordance exists between the measured and predicted values of PPV and AOp. This is because of the capability of PSO to minimize the error function with high efficiency; the PSO algorithm adjusts the weights and biases of the error objective function of ANN to obtain the minimum MSE. The same results were obtained for the testing datasets, as can be seen in Fig. 14. As shown in the figure, the predicted values of PPV and AOp obtained by employing the proposed PSO-based ANN model are in close agreement with the measured values. It is worth noting that the results of the optimum PSO-based ANN model presented in these figures are the best obtained by model no. 3, whereas the results tabulated in Table 5 are the average values of five repeated runs.

Fig. 13
figure 13

Comparison between measured and predicted a PPV and b AOp for different training datasets

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Fig. 14
figure 14

Comparison between measured and predicted a PPV and b AOp for different testing datasets

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PPV and AOp prediction through PSO-based ANN model and empirical approaches

A comparison was conducted between the values of PPV and AOP predicted by the proposed PSO-based ANN model, empirical approaches, and measured values to check the accuracy of the proposed model. For this purpose, ten datasets were selected in terms of the distance between the free face and the monitoring point, rock properties, and blasting parameters, as listed in Table 6.

Table 6 Investigated parameters to estimate PPV and AOp by PSO-based ANN model and empirical approaches
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Figure 15 shows a comparison between the measured PPV and predicted PPV values obtained by empirical approaches (see Table 1) and the proposed PSO-based ANN model. It can be seen that the PPV values obtained by the PSO-based ANN model are in very good agreement with the measured values, whereas there are wide variations in PPV values predicted by empirical methods.

Fig. 15
figure 15

Comparison of PPV for selected datasets

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For the selected datasets, AOp values were obtained by means of empirical approaches (Eqs. 1, 2, and 4). Based on the site conditions, three different sets of site factors (H = 22,182 and β = 0.966; H = 19,062 and β = 1.1; H = 261.54 and β = 0.706) were extracted from Table 2. Subsequently, a comparison was made between the measured AOp values and predicted AOp values obtained by the PSO-based ANN model and empirical approaches, as shown in Fig. 16. According to the figure, the proposed PSO-based ANN model yields more accurate results compared to empirical approaches. Based on Figs. 15 and 16, it can be concluded that the proposed PSO-based ANN model is an applicable tool for the prediction of PPV and AOp induced by blasting of this quarry with a high degree of accuracy.

Fig. 16
figure 16

Comparison of AOp for selected datasets

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Sensitivity analysis

A sensitivity analysis was carried out to identify the relative influence of each parameter in the neural network system by the cosine amplitude method (Yang and Zang 1997). To apply this method, all data pairs were expressed in common X-space. The data pairs used to construct a data array X are defined as:

$$X \, = \, \left\{ {x_{1} , \, x_{2} , \, x_{3} \ldots , \, x_{i} , \, \ldots , \, x_{n} } \right\}.$$

The elements x i in the array X are a vector of length m, that is:

$$x_{i} = \, \left\{ {x_{i1} , \, x_{i2} , \, x_{i3} \ldots , \, x_{im} } \right\}.$$

Each of these data pairs can be trained as a point in m-dimensional space, where each point requires m-coordinates for a full description. Thus, in the space pair, all the points are associated with the achieved results. The following equation illustrates the strength of the relation (r ij ) between the dataset X i and X j :

$$r_{ij} = \frac{{\mathop \sum \nolimits_{k = 1}^{m} x_{ik} x_{jk} }}{{\sqrt {\mathop \sum \nolimits_{k = 1}^{m} x_{ik}^{2} \mathop \sum \nolimits_{k = 1}^{m} x_{jk}^{2} } }}.$$
(11)

Table 7 shows the strengths of the relations (r ij values) between the input and output (PPV and AOp) parameters. The sensitivity analysis results show that sub-drilling (E) and maximum charge per delay (B) are the parameters with the greatest influence on PPV, whereas stemming length (D) and maximum charge per delay (B) are the parameters with the greatest influence on AOp.

Table 7 Strengths of relation between input and output parameters
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Conclusion

A MatLab code was developed to predict blast-induced PPV and AOp using a hybrid PSO-based ANN model. Eighty-eight datasets collected from Hulu Langat granite quarry site in Malaysia were used to develop an optimum PSO-based ANN model. Hole depth, maximum charge per delay, burden-to-spacing ratio, stemming length, sub-drilling, powder factor, RQD, distance between the free face and the monitoring point, and number of holes were used as input parameters, while PPV and AOp values were set as output parameters. A series of sensitivity analyses were conducted to determine the optimum PSO parameters. The optimum network architecture was determined following the trial and error method. Finally, a model with one hidden layer and 12 nodes in the hidden layer was selected to be used for prediction. A comparison was made between the results obtained by the PSO-based ANN model and empirical predictors as well as the measured values to examine the applicability and accuracy of the proposed model. The results indicate that the proposed PSO-based ANN model is practically able to predict PPV and AOp induced by blasting in granite quarry sites with similar conditions. Through the sensitivity analyses, it was also found that the sub-drilling and maximum charge per delay are the parameters with the greatest influence on PPV, whereas the stemming length and maximum charge per delay are the parameters with the greatest influence on AOp.