Development of an ANN-Based Closed-Form Equation for the Prediction of Airblast Overpressure Induced by Construction Rock Excavation Blasting in Urban Areas

Abstract

Blasting has been proven to be the most cost-effective method for rock excavation known to man. The cost-effectiveness advantage of blasting is overshadowed by its unpleasant environmental problems, particularly at construction sites close to human settlements and public utilities. Therefore, efforts are required to develop closed-form equations that can accurately predict environmental problems associated with blasting. This study proposes an ANN-based closed-form explicit equation for forecasting airblast overpressure (AOp) at multiple construction sites in South Korea. Nine important factors that affect AOp generation were used to develop the model. First, a stand-alone ANN was initiated, and the hyperparameters of the optimum ANN structure were tuned using two novel and robust metaheuristic algorithms: the slime mould algorithm (SMA) and multi-verse optimization (MVO). To appraise the predictive accuracy of the developed soft computing models, multilinear regression (MLR) and a generalized empirical predictor were developed for comparison. The analysis showed that the SMA-ANN and MVO-ANN models predicted AOp with the highest accuracy compared with the other models. The two hybrid ANN-based models were transformed into closed-form and explicit equations to aid in the easy forecasting of AOp when planning a blasting round at construction sites. The developed model equations were validated for practical engineering applications and a comprehensive relative importance analysis of the AOp input parameters was performed. The relevance importance analysis shows that the rock mass rating (RMR), charge per delay (Q), and monitoring distance (DIS) have the highest impacts on AOp.

1. Introduction

Rapid industrialization and urbanization have increased the demand for new and improved infrastructural amenities that have continued to rise in recent years, owing to the continuous and astronomical surge in human population, which has overstretched the existing constructed infrastructure worldwide. Rock excavation by blasting has been widely and successfully used in recent years in civil construction works, including urban land and redevelopment construction projects, dam and hydropower projects, tunneling and underground spaces, and mineral exploitation, because of its affordability, applicability to any geological terrain and lithologies, and the production of reliable results in comparison with other methods [1,2,3,4,5]. However, some percentage of the confined explosive energy that is meant for rock fragmentation during blasting escapes into the surrounding space, causing various unwanted environmental events, including air blast or air overpressure (AOp), blast-induced ground vibration (BIGV), flyrock, overbreak, and noxious gases (Figure 1) [6,7,8]. These unwanted environmental events from blasting must be addressed seriously, particularly in construction projects where blasting is undertaken near human settlements and public utilities. Hence, the accurate forecasting of the severity level of these events at varying distances from the point of blast initiation is crucial to prevent and minimize their occurrence and, subsequently, to ensure the effective use of rock excavation by blasting in construction projects.

Airblast overpressure (AOp), also known as the “overpressure wave,” is an unwanted event that usually emanates from blasting operations and has recently received considerable attention owing to its virulence. AOp can cause deleterious damage to nearby buildings and important utilities and greater harm to nearby inhabitants. Some of the adverse effects of virulent AOps include the shattering of windows and roofing, visible structural cracking, deafness, the irritation of nearby residents, and eventual unresolved issues and litigation between inhabitants, construction companies, and mines [9,10]. These issues could become even more critical in urban areas where blasting is performed near residential buildings and public infrastructure [4]. Owing to its potential to harm inhabitants and damage important structures and utilities, researchers have attempted to develop methods for quantifying AOp generation during blasting. These methods were developed using the influential factors causing AOp generation, including drill and blast design (controllable factors), meteorological factors, and geological factors (uncontrollable factors). The controllable factors include the burden, spacing, charge weight, monitoring distance, and explosive parameters, whereas the uncontrollable parameters include the rock and rock mass strength, such as the uniaxial compressive strength, and meteorological parameters, such as temperature and wind direction. Researchers have proposed empirical models and equations that use two of the blast design parameters, explosive maximum charge weight and the distance between the point of blast initiation and seismograph position, with some level of accuracy [11,12]. These empirical relations are easier to employ, less laborious, and more cost-effective. However, owing to geological heterogeneity and differences in blast area conditions from location to location, these models are generally mine-specific and often less accurate [13,14]. Compared to empirical equations, numerical models provide more accurate models that are closer to field measurements, as demonstrated by Wu and Hao [15] and Zhou et al. [16] using Autodyn2D and LS-Dyna, respectively. However, numerical solutions are often time-consuming and costly. Hence, there is a need to devise a more reliable and accurate, albeit non-time-consuming, method for quantifying the hazardous impacts of AOp using soft computing or artificial intelligence methods.

Soft computing techniques such as artificial neural networks (ANNs), random forest (RF), ensemble methods, extreme learning machines (ELMs), fuzzy inference systems, support vector machines (SVMs), and hybrid models have recently been widely used to forecast AOp [7,14,17,18,19,20]. The surge in the use of these intelligent methods can be attributed to the accuracy, cost-effectiveness, and non-time consumption of these models. Soft computing methods are versatile approximation methods owing to their non-parametric nature and ability to accept additional factors that influence AOp generation [7,21]. Zeng et al. [19] forecasted AOp due to blasting using 62 blasting events measured at quarry sites in Malaysia using a cascaded forward neural network (CFNN) trained by the Levenberg–Marquardt (LM) algorithm (CFNN-LM) and compared it with the results of a generalized regression neural network (GRNN) and an extreme learning machine (ELM). They showed that the forecasted values of AOp by the CFNN-LM were much closer to the field measurements than those by the GRNN and ELM. In another study, Khandelwal and Kankar [14] employed an SVM model to predict AOp using 75 datasets from three mines in India and compared its predictive performance with that of an empirical equation. The analyses showed that the SVM model was better than the empirical equation for predicting AOp. Nguyen and Bui [17] predicted the AOp using hybridized ANN-RF, ANN, and RF models from 114 blasting events obtained from a quarry in Vietnam. Based on a comparison of these methods with empirical models, they concluded that soft computing models provide more accurate predictions than empirical models. Similar studies on the utilization of soft computing models to predict AOp can be found in [8,22,23,24,25,26,27,28,29,30,31,32,33,34,35]. According to the no-free-launch (NFL) hypothesis, despite the advantages of various machine learning (ML) algorithms, no single ML technique can be considered the most appropriate method for solving all engineering problems [36]. Thus, numerous ML algorithms have been investigated, and new algorithms are being developed to solve various complex engineering optimization problems, including AOp prediction. Furthermore, the existing AOp models are limited to one or a few sites and are all based on blasting for mine production, and none accurately predict the AOp generated by blasting for construction rock excavations.

Figure 1. Blastwave propagation and its unwanted environmental events (adapted from [13]).

Figure 1. Blastwave propagation and its unwanted environmental events (adapted from [13]).

Thus, this study proposes novel optimization algorithms known as the slime mold algorithm (SMA) and multi-verse optimization (MVO) to optimize an ANN and predict AOp generation from multiple construction blasting sites. To the best of our knowledge, the SMA has never been used for the prediction of AOp in the literature. The first novelty of this study is the transformation of AOp ANN-based models into easy-to-implement, explicit, and closed-form equations. The developed explicit, closed-form equations can be easily implemented in an Excel spreadsheet without prior knowledge of ANN principles, thereby simplifying the practical engineering implementation of the models. The second novelty lies in the employment and evaluation of the impacts of both controllable and uncontrollable factors (rock mass rating) on AOp generation. Because most existing AOp models rely solely on the drill and blast parameters, incorporating uncontrollable factors makes the developed models more robust. The final innovation in this study is the practical engineering validation of the developed models with randomly selected datasets. The outcomes of the SMA-ANN and MVO-ANN models were compared with those of the multilinear regression (MLR) and empirical equations. The findings of this study will help predict AOp generation at construction sites, particularly during the design and pre-construction phases.

2. Data Collections and Method

2.1. Description of the Study Sites and Data Collection Method

The study sites were highway and urban land development and redevelopment construction project sites in South Korea, as shown in Figure 2b. The datasets used for this study were collected from blasting operations during the pre-construction phase of these highways and urban land and redevelopment projects. The geology of South Korea is characterized by three main massifs: the Precambrian Nangrim, Gyeonggi, and Yeongnam massifs. The Gyeonggi and Yeongnam massifs are composed of basement Precambrian gneiss complexes consisting of 2.7 to 1.1 Ga high-grade gneiss and schists. The two massifs are separated by the Okcheon fold belt, which comprises metasedimentary rocks and bimodal meta-volcanic rocks, such as metapelites, quartzites, conglomeratic phyllites, marble, and calc-silicates, as shown in Figure 2a [37,38]. However, the rock exposures and bedrock found at the study sites included granite, gneiss, shale, tuff, sandstone, andesite, schist, and rhyolite. Following [39,40,41,42]’s classification methods, the rock weathering degrees observed in all the sites under study were classified as grades II to IV. Three seismographs, Blastmate II and III, and Minimate seismographs (Instatel Corporation Canada), each outfitted with two accessories (triaxial geophones and microphones), were used to measure the AOp (dB) in these construction sites. These devices recorded AOp values that ranged from 100 dB (2 Pa) to 148 dB (256 Pa). The operating frequency response of the microphones was 2–300 Hz, which was sufficient to accurately measure the AOp in the frequency range critical for important utilities, structures, and human hearing. Before the blast, two seismograph accessories were connected to the seismographs and were placed at predetermined points in the blast area. The monitoring distances of the AOp ranged from 14 m to 1344 m and were not the same for each site. In total, 115 blasting events were recorded at these sites. The dataset comprised the hole diameter (D), blasthole depth (L), powder factor (PF), charge per delay (Q), spacing (S), stemming length (T), burden (B), rock mass rating (RMR), monitoring distance (DIS), and AOp (dB) for each blast. The explosives used in all blasting were 32 mm- and 50 mm-diameter emulsion explosives with a millisecond delay electronic detonator. Stemming materials included sand, gravel, and rock dust. The diameters of the blastholes at all sites were 51 and 76 mm, respectively. The distance between the AOp monitoring point and blast initiation was measured using a handheld global positioning system (GPS). The RMR values were obtained from experiments conducted on core logs from each site. A breakdown of the dataset statistics used in this study is presented in

Table 1. Brief statistical description of AOp model training and test datasets.

Data	Statistics	B (m)	S (m)	D (mm)	L (m)	T (m)	PF (kg/m3)	Q (kg)	DIS (m)	RMR	AOp (dB)
Training	Count	81	81	81	81	81	81	81	81	81	81
	Mean	0.99	1.22	60.57	2.61	0.69	0.39	1.47	190.09	57.79	67.64
	Min	0.40	0.40	51	1.20	0.28	0.21	0.10	14	40	51.5
	Max	2	2.50	76	6.20	1.40	0.78	8.00	1120	73	81.9
	Std	0.42	0.55	12.23	1.04	0.29	0.12	1.65	179.45	7.36	7.52
Test	Count	34	34	34	34	34	34	34	34	34	34
	Mean	0.91	1.14	56.15	2.53	0.64	0.40	1.24	213.88	58.53	67.37
	Min	0.40	0.40	51	1.2	0.28	0.27	0.13	37	44	53.8
	Max	2	2.5	76	6.2	1.40	0.78	8.00	1344	73	84.6
	Std	0.35	0.48	10.26	0.99	0.25	0.11	1.62	280.41	7.62	7.46

. Figure 3 shows the correlation plots between the model input variables and between the inputs and outputs.

3. Materials and Methods

3.1. AOp Prediction Model Development

The implementation of the predictive models developed for the prediction of AOp is described in this section. The models were implemented by employing 115 datasets obtained during the design and pre-construction phases of highways, urban land, and redevelopment projects in South Korea. As discussed in Section 2.1, the nine predictor parameters used to develop the models include the hole diameter (D), blasthole depth (L), powder factor (PF), charge per delay (Q), spacing (S), stemming length (T), burden (B), rock mass rating (RMR), and monitoring distance (DIS), and AOp (dB) is the predicted parameter.

3.1.1. ANN

The ANN was chosen as the base model to develop the forecasting model in this study, not only because of its wider acceptance for predicting AOp and other deleterious environmental products of blasting but also because it offers simple implementation and flexibility in depicting input–output relationships. ANN modeling was initiated in MATLAB after the initial preprocessing and normalization of the datasets between 1 and −1 using Equation (1) [43]:

$${\omega }_{norm}=\frac{2(\omega -{\omega }_{\mathit{min}})}{({\omega }_{\mathit{max}}-{\omega }_{\mathit{min}})}-1$$

(1)

where ω_norm denotes the scaled variable; ω represents the field measurement to be scaled; and ω_min and ω_max are the values of the lowest and highest input and output parameters in the real dataset, respectively.

The preprocessed or normalized data were then randomly divided into training/validation and test datasets using 70% and 30% division ranges, respectively. The training datasets were then downloaded into MATLAB for training, and the test datasets were later used to appraise the performance of the trained datasets on the unseen datasets. An ANN structure with three layers, that is, input, hidden, and output layers, was employed in this study. The datasets entered the developed network through the input neurons and were further processed in the hidden neurons before coming out as predicted outcomes in the output neurons. The hyperbolic tan-sigmoid activation function was applied to the input parameters during training. The Levenberg–Marquardt training function was used as the feed-forward algorithm for the ANN model, and the mean square error (MSE) was chosen as the performance objective function.

The hidden layer neurons were varied using several combinations, and the coefficients of correlation (R) for each tested hidden neuron are listed in

Table 2. Tried ANN architectures.

ANN	Training	Testing	Validation	Overall
9-2-1	0.71817	0.58937	0.85463	0.71575
9-3-1	0.72639	0.80024	0.88343	0.75687
9-4-1	0.83252	0.89655	0.66851	0.81560
9-5-1	0.79340	0.76551	0.82454	0.79199
9-6-1	0.82267	0.85674	0.73655	0.81916
9-7-1	0.83482	0.90343	0.89298	0.84807
9-8-1	0.80689	0.93599	0.66849	0.80711
9-9-1	0.88673	0.98163	0.73014	0.88623
9-10-1	0.78720	0.83888	0.87414	0.80637
9-11-1	0.83647	0.86453	0.94504	0.85692
9-12-1	0.82565	0.74465	0.74328	0.80500

. From Table 2, the ANN with nine neurons in the hidden layer (9-9-1) adjudges the optimum network based on the coefficient of correlation (Figure 4). However, to obtain a model that can satisfactorily replicate actual field measurements, the chosen optimum ANN architecture performance in Table 2 still needs to be improved. Consequently, the accuracy of the chosen optimum ANN is enhanced by means of the novel optimization algorithms described in the next section.

3.1.2. Optimization of ANN Model

The optimum ANN was optimized to further improve its predictive performance. Optimization algorithms improve the performance of ANNs by solving the problems of local minima and slower convergence speeds. The use of optimization algorithms is also cost-effective, as it prevents the use of many neurons in the hidden layer before achieving an accurate model. In this study, a nature-inspired slime mould algorithm (SMA) and a physics-based nature-inspired multi-verse optimizer (MVO) were employed to tune the optimum ANN structure hyperparameters. The selection of these two algorithms to enhance the predictive performance of the ANN in this study was based on their robustness and the need to appraise their predictive performance in predicting AOp. The processes used to execute the optimized ANN models proposed in this study are discussed below.

The SMA is modeled based on the oscillation pattern of a eukaryotic organism known as slime mold (SM) in its natural environment. The algorithm exhibits a unique mathematical model that replicates the foraging behavior of SM when forming a route in search of food and survival in its natural habitat [44]. The model replicates the mode of generation of positive and negative feedback exhibited by the SM during the foraging process (Figure 5). It features a flexible framework that consistently maintains a balance between local and global search drifts. The two phases involved in the SMA’s implementation are “approaching food” and “warping food,” as discussed below.

• Approaching food

In this initial phase, the SM acquires food based on its odor in the air. The approaching behavior of SM can be described mathematically by Equation (2):

$$\overrightarrow{M(l+1)}=\left\{\overrightarrow{M_b(l)}+\overrightarrow{V_b}\right.\ast (\overrightarrow{W}\ast \overrightarrow{M_A(l)}-\overrightarrow{M_B(l)},r<p\overrightarrow{V_c}\ast \overrightarrow{M(l)},r\ge p$$

(2)

where $\overrightarrow{V_b}$ is a variable which has values between −a and a; $\overrightarrow{V_c}$ linearly decreases from 1 to 0. l represents the current iteration, $\overrightarrow{M_b}$ describes each position with optimum odor concentration, $\overrightarrow{M}$ describes the location of the SM in nature, $\overrightarrow{M_A}$ and $\overrightarrow{M_B}$ are two members that were randomly picked from the group, and $\overrightarrow{W}$ denotes the SM’s body weight. The parameter $‘p’$ in Equation (2) can be expressed as shown in Equation (3):

$$p=\tanh \left|G(i)-HF\right|$$

(3)

where G(i) denotes the fitness of $\overrightarrow{M}$, HF is the optimal solution found in all iterations, and i ∈ 1, 2,…,n. The mathematical equations for $\overrightarrow{V_b}$ and the SM’s body weight, $\overrightarrow{W}$, are as expressed in Equations (4)–(7):

$$\overrightarrow{V_b}=\left[-a,a\right]$$

(4)

$$a=\arctan h(-\left(\frac{l}{\mathit{max}\_l}\right)+1)$$

(5)

$$\overrightarrow{W(smellIndex(i))}=\left\{\begin{array}{l}1+r\ast \log \left(\frac{bF-G(i)}{bF-wF}+1\right),condition\\ 1-r\ast \log \left(\frac{bF-G(i)}{bF-wF}+1\right),others\end{array}\right.$$

(6)

$$SmellIndex=sort(G)$$

(7)

b.

Warping food

This second phase depicts the contraction behavior of the intravenous network of slime mould while hunting for food in its natural habitat. This phase can be mathematically expressed as Equation (8):

$$\overrightarrow{M^{\ast }}=\left\{\begin{array}{l}rand\ast (ob-mb)+mb,rand<z\\ \overrightarrow{M_b(l)}+\overrightarrow{V_b}\ast (\overrightarrow{W}\ast \overrightarrow{M_A(l)}-\overrightarrow{M_B(l)},r<p\overrightarrow{V_c}\ast \overrightarrow{M(l)},r\ge p\end{array}\right.$$

(8)

where mb and ob describe the lowest and highest limits of the search area, and rand and r represent the area estimate or value between 0 and 1.

The pseudocode for SMA model implementation is available in Li et al. [44].

The MVO, as introduced by Mirjalili et al. [45], is a nature-based optimization algorithm, like the SMA, but it differs from the SMA in that it is based on the multi-verse theory in physics. The optimizer consists of three distinct phases, exploration, exploitation, and local search, based on the main concepts and mathematical structures of black, white, and wormholes (Figure 6a) [46]. Similar to other population-based metaheuristic algorithms, the search process is divided into exploration and exploitation. The ideas and theories of white and black holes are employed to explore search spaces, whereas wormholes help the optimizer in search space exploitation [46]. In the MVO algorithm, universes are considered solutions and are expressed by the size of their populations. In addition, each object in the universe is specified as a parameter in the search area of the universe. The inflation rate is used to appraise the fitness of each solution while the objects move in the search area (Figure 6b).

The optimizer applies this set of guidelines during the optimization process:

1.

There is a higher positive proportionality between the inflation rate and white holes.

2.

There is a lower negative proportionality between the inflation rate and black holes.

3.

Universes with higher inflation rates are liable to transmit objects with the assistance of white holes.

4.

Universes with lower inflation rates are liable to accept multiple objects with the assistance of black holes.

5.

The objects in the universes may experience periodic motion towards the solution space through wormholes, irrespective of the rate of inflation.

During the exploration stage, a roulette wheel mechanism was chosen to simulate the white- and black-hole tunnels computationally and exchange the objects of the solutions. The solutions were then sorted based on their inflation rates at each iteration, and a solution was selected using a roulette wheel to obtain a white hole. Taking G as the entire universe (solution), as in Equations (9) and (10), respectively:

$$G=\left(\begin{array}{ccc}{y}_1{}^1& y_1{}^2\dots & y_1{{}^d}_{}\\ ⋮& \ddots & ⋮\\ y_n{}^1& y^2{}_n\cdots & y^d{}_n\end{array}\right)$$

(9)

$$y_i^j=\left\{\begin{array}{l}{y}_k^{j}r1<NI(Gi)\\ y{}_i{}^{j}r1\ge NI(Gi)\end{array}\right.$$

(10)

where d is the number of parameters, n denotes the number of solutions (universe), yji denotes the jth parameter of the ith solution (universe), Gi denotes the ith solution (universe), NI (Gi) denotes the ith universe’s normalized inflation rate, r1 is a uniformly distributed random number in the range (0, 1), and yjk denotes the jth parameter of the kth universe as chosen by the roulette wheel method.

In the exploitation phase, wormhole tunnels are created using a solution (a universe), and the optimal solution formed thus far ensures local changes for each solution and has a greater tendency to enhance the rate of inflation through wormholes. This method is described by Equations (11) and (13), as follows:

$$\begin{array}{l}{y}_i^j=\left\{\begin{array}{l}{y}_j+TDR\times ((g{b}_j-l{b}_j)\times r4+l{b}_j)r3<0.5\\ y_j-TDR\times ((g{b}_j-l{b}_j)\times r4+l{b}_j)r3\ge 0.5\end{array}\right.r2<WEP\\ r2\ge WEP\end{array}$$

(11)

where y_j is the jth variable of the optimal solution formed thus far; lb_j is the jth variable’s minimum value; ubj is the maximum value of the jth variable; xji is the jth variable of the ith solution; and r2, r3, and r4 are randomly distributed random numbers in the range (0, 1).

The wormhole existence probability (WEP) determines the likelihood of finding a wormhole in a solution. The traveling distance rate (TDR) is a variable that describes the rate (variation) at which an object can be teleported using a wormhole throughout the best-fit universe discovered thus far.

$$WEP=\mathit{min}+l\times \left(\frac{\mathit{max}-\mathit{min}}{L}\right)$$

(12)

$$TDR=1-\frac{l^{1/p}}{L^{1/P}}$$

(13)

where “min” is the lower limit, “max” is the upper limit, “l” denotes the present iteration, “L” denotes the optimal number of iterations, and “p” denotes the exploitation reliability over the iterations.

The SMA and MVO algorithms were executed in MATLAB using written code. The optimum ANN architecture (9-9-1) in Section 2.1 was initialized through training and testing and then subjected to hyperparameter (weights and biases) optimization using the SMA and MVO algorithms for the SMA-ANN and MVO-ANN models, respectively. The number of search agents in both SMA and MVO was adjusted from 10 to 30 while maintaining the maximum number of iterations at 1000. Section 3 discusses the predictive performance of the developed SMA-ANN and MVO-ANN models for AOp prediction from construction rock excavation blasting.

3.1.3. Multilinear Regression (MLR) Model

MLR is a technique commonly used to illustrate the linear relationship between one or more independent variables and a single dependent variable. This relationship is based on the premise of reducing the sum of squares of the distinctions between the predictor and predicted variables [47]. The general equation of MLR can be written as shown in Equation (14):

$$Z=c_o+c_1{y}_1+c_2{y}_2{c}_3{y}_3+.........c_n{y}_n+\mu$$

(14)

where Z and y_i to y_n are the predicted and predictor variables, respectively; c_o is the intercept; $c_1$ to $c_n$ are the coefficients of the predictor variables; and μ is the predictor-associated error.

The purpose of the developed MLR model is to compare the developed soft computing and empirical models with those of the MLR-generated model for AOp prediction. The same sets of data used to generate the soft computing models were used to develop the MLR equations. However, the datasets were not normalized before the MLR simulation was performed. MLR modeling was performed using OriginPro 2021 software. Equation (15) shows the generated MLR equation for AOp prediction.

$$AOp=75.3177-10.5074B+4.9811S-0.0308D+2.1664L-6.7785PF+0.6264Q-0.0108DIS-0.0587RMR$$

(15)

3.1.4. Empirical Predictor

Empirical equations have been developed by several blasting regulatory institutes to predict the AOp in the absence of field data. In this study, the United States Bureau of Mines (USBM) empirical predictor proposed by Siskind et al. [12] was used to predict AOp generation from construction sites. The aim was to compare the empirical predictor results with those of the soft computing models. The AOp empirical predictor equation is expressed in Equation (16).

$$AO{p}_{USBM}=\alpha \times {\left[\frac{DIS}{\sqrt[3]{Q}}\right]}^{-\mu }$$

(16)

where $\alpha$ and μ are the site constants obtained from multivariate regression analysis by plotting the training sets’ field-measured AOp against the scaled distance on a log–log graph using the training datasets in Origin software (Figure 7). The Levenberg–Marquardt iteration method and an allometric function for nonlinear curve fitting were used. Equation (16) can then be expressed in its final form to make predictions on the training and testing datasets, as represented in Equation (17).

$$AO{p}_{USBM}=94.184\times {\left[\frac{DIS}{\sqrt[3]{Q}}\right]}^{-0.067}$$

(17)

4. Appraising Model Prediction Performance

The predictive performance of the developed models for AOp prediction in construction rock excavation blasting was appraised using established statistical error metrics, such as mean square error (MSE), root mean square error (RMSE), mean absolute error (MAE), relative root mean square error (RRMSE), mean absolute percentage error (MAPE), and coefficient of determination (R²). The mathematical representations of these error metrics are given by Equations (18)–(23).

$$R^2=1-\frac{{\displaystyle \sum {}_i{(AOp-AO{p}_i)}^2}}{{\displaystyle \sum {}_i{(AO{p}_i-\overline{AOp})}^2}}$$

(18)

$$MAE=\frac{1}{n}{\displaystyle \sum _{i=1}^n\left|AOp-AO{p}_i\right|}$$

(19)

$$MSE=\frac{1}{n}{\displaystyle \sum _{i=1}^n{(AOp-AO{p}_i)}^2}$$

(20)

$$RMSE=\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^n{(AOp-AO{p}_i)}^2}}$$

(21)

$$RRMSE=\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^n\sqrt{{\left(\frac{AOp-AO{p}_i}{AOp}\right)}^2}}}$$

(22)

$$MAPE=\frac{100\%}{n}{\displaystyle \sum _{i=1}^n\left|\frac{AOp-AO{p}_i}{AOp}\right|}$$

(23)

where AOp represents the observed (actual) values (dB), AOp_i represents the values forecasted by the developed models (dB), $\overline{AOp}$ represents the average value of the observed (actual) values (dB), and n is the total data points in the employed datasets.

5. Results and Discussion

5.1. AOp Model Performance Appraisal Results and Discussion

The predictive performances of the AOp prediction models generated in this study were appraised using the error metrics listed, and the results are presented in

Table 3. AOp prediction model performance appraisal results.

Stage	AOp Model	R2	MAE	MSE	RMSE	MAPE	RRMSE
Training	ANN	0.786	2.881	12.820	3.581	4.350	0.056
	SMA-ANN	0.809	2.363	10.803	3.286	3.576	0.051
	MVO-ANN	0.805	2.441	11.712	3.422	3.740	0.053
	MLR	0.122	5.768	49.115	7.008	8.744	0.110
	USBM	0.226	5.272	43.305	6.581	8.011	0.103
Test	ANN	0.788	2.303	10.319	3.212	3.418	0.048
	SMA-ANN	0.867	2.102	8.084	2.843	3.245	0.045
	MVO-ANN	0.830	2.254	8.323	2.885	3.380	0.047
	MLR	0.044	6.555	77.281	8.791	9.800	0.128
	USBM	0.049	6.084	63.824	7.989	9.118	0.117

for the training and test data, respectively. From Table 3, it is evident that soft computing models such as ANN, SMA-ANN, and MVO-ANN perform better than the empirical predictor and MLR models. Starting with the stand-alone ANN model, the values of MAE, MSE, RMSE, MAPE, RRMSE, and R² were 2.881, 12.820, 3.581, 4.350, 0.056, and 0.786 for the training datasets, and 2.303, 10.319, 3.212, 3.418, 0.048, and 0.788 for the test datasets, respectively. The developed SMA-ANN model displayed the smallest MAE, MSE, RMSE, MAPE, and RRMSE and the largest R² values of 2.363, 10.803, 3.286, 3.576, 0.051, and 0.809 for the training datasets and 2.102, 8.084, 2.843, 3.245, 0.045, and 0.867 for the test datasets, respectively. For the MVO-ANN model, the MAE, MSE, RMSE, MAPE, RRMSE, and R² values for the training datasets are 2.441, 11.712, 3.422, 3.740, 0.053, and 0.805, and 2.254, 8.323, 2.885, 3.380, 0.047, and 0.830, for the test datasets, respectively. The MLR model had the highest error metrics and the poorest performance of all the developed models, with MAE, MSE, RMSE, MAPE, RRMSE, and R² values of 5.768, 49.115, 7.008, 8.744, 0.110, and 0.122 for the training datasets and 6.555, 77.281, 8.791, 9.800, 0.128, and 0.044 for the test datasets, respectively. The performance of the USBM empirical predictor commonly used in rock construction excavation blasting is averagely poor, with MAE, MSE, RMSE, MAPE, RRMSE, and R² values of 5.272, 43.305, 6.581, 8.011, 0.103, and 0.226 for the training datasets and 6.084, 63.824, 7.989, 9.118, 0.117, and 0.049 for the test datasets, respectively.

The findings in Table 3 are supported by a graphical representation of the relationship between field measurements and forecasted AOp by the generated models, as shown in Figure 8a–e. The SMA-ANN model fit line was determined to be closer to the perfect fit line, with R2 values close to the perfect R² values for both the training and test datasets.

5.2. AOp Prediction Model Comparison

As a further means of illustrating the overall performance of the optimum SMA-ANN model and to make comparisons with other developed soft computing, MLR, and empirical models, a Taylor diagram (Figure 9) was made using the overall datasets instead of the training and test datasets in Table 3 and Figure 8, respectively. The Taylor diagram combines three error metrics—namely, the coefficients of correlation (R), standard deviation (Std), and root mean squared difference (RMSD)—to evaluate and compare model performance [48]. As shown in Figure 9, the SMA-ANN model had the lowest RMSD and highest R values compared with the MVO-ANN, stand-alone ANN, MLR, and USBM models. This shows that the SMA-ANN model is the most robust of all developed models for predicting AOp during rock excavation blasting at construction sites. The performance of the MVO-ANN model was close to that of the SMA-ANN model, with little observed difference in RMSD and R values compared to the SMA-ANN model against the reference or observed point (blue color). Compared to the MLR and USBM models, the three soft computing models (SMA-ANN, MVO-ANN, and ANN) were superior in predicting AOp from construction rock excavation blasting, judging from their lower RMSD and higher R values. The poor performance of both the MLR and USBM models could be due to their inability to capture the nonlinear relationships that exist between the AOp and the nine input parameters. In addition, it can be concluded that the use of two metaheuristic algorithms to tune the ANN hyperparameters improves the accuracy and predictive performance of the hybrid ANN models.

To further compare the developed AOp models, all six error metrics were compared for the test datasets, as shown in Figure 9b. The comparison results show that the optimum SMA-ANN model when using a Taylor diagram is also the most accurate and robust of all the developed models. This is closely followed by the MVO-ANN, stand-alone ANN, MLR, and USBM models. Thus, the SMA-ANN and MVO-ANN models are excellent and are proposed for the prediction of AOp at construction sites.

5.3. Development of Closed-Form AOp ANN-Aided Prediction Equations

The retrieved network weights and biases of the two optimized ANN models (SMA-ANN and MVO-ANN) are listed in

Table 4. Extracted weights and biases of the two optimized ANN models.

Model	Weights										Biases
	B	S	D	L	T	PF	Q	DIS	RMR	PPV	b1	b0
SMA-ANN	0.9218	−0.2065	2.0207	−2.1869	0.9217	4.2406	3.4291	0.0233	−2.1843	−5.2400	−1.7374	1.4006
	−0.1981	1.2497	−1.1091	−3.1843	−0.1999	−1.4348	−0.4071	0.1439	2.1539	−0.6904	−3.5815
	2.9352	−0.5492	−2.0234	−0.2689	4.7197	2.0032	0.5673	2.1550	−1.7423	3.3269	0.5541
	2.8495	−5.3477	−2.1881	2.2159	4.8837	1.1803	−5.0623	5.5159	−0.1145	−3.7788	3.0744
	1.9090	−4.3624	−0.0267	12.0182	1.8981	−3.2526	−4.5959	6.2092	−1.3156	3.3778	3.9849
	−1.3518	1.6233	−0.2834	−2.7341	−1.3456	0.1097	0.5537	−3.7619	0.3139	8.3641	−3.8070
	1.7754	−1.8998	−0.0859	3.5352	1.8467	0.4503	−7.0705	2.9821	0.5262	4.0458	−1.5019
	0.4930	0.1933	−0.0307	−7.3747	0.2709	−0.0737	−6.3318	0.1578	−0.1736	−0.2037	−0.0216
	1.1422	10.6302	−1.0916	−3.1519	1.0553	2.7472	−0.9890	4.0753	0.1246	1.1364	4.9380
MVO-ANN	3.2593	7.1832	−2.4079	−0.1395	3.9862	−5.9387	−0.9281	3.3369	4.4127	0.6818	3.6026	1.3982
	−1.6501	−1.1915	−0.1357	−0.1925	−0.9058	−0.1382	−1.9574	−1.0608	0.8722	4.7120	2.7622
	−1.0330	−0.6646	−0.2764	−3.3653	−1.1693	−2.7454	−1.7060	2.2366	−0.7899	−1.4915	−1.8593
	0.1325	−2.7229	0.7861	0.7534	−0.5904	2.8602	−2.0336	−1.0599	−3.9818	−1.7079	1.4799
	2.1845	−0.1520	1.0671	−0.5641	1.9426	0.4680	0.9504	3.7224	−0.0898	2.8986	−1.4936
	0.8432	2.5071	0.8191	0.3777	−0.9472	−2.9196	−0.1334	4.8667	6.8262	−1.6146	−1.4919
	−0.0580	3.7126	−0.1745	−3.3070	−0.8605	0.9504	1.4986	−3.7591	0.7592	−2.9749	−2.5205
	−2.9660	−3.6424	3.3060	0.3393	−2.3650	9.7145	1.8197	0.6470	1.3214	0.8277	3.6160
	1.0084	−2.3431	0.4299	1.1599	0.8472	−0.1035	−1.2063	3.8228	−0.0846	−4.7781	3.5630

. The weights and biases were successfully converted into simple and adaptable closed-form explicit equations that can be used to predict AOp generation when planning a blasting round for construction rock excavations. The generated AOp predictor equation for the SMA-ANN model is presented in Equation (24).

$$AO{p}_{SMA-ANN}=16.55tanh\left({\displaystyle \sum _{i=1}^9{x}_i+1.4006}\right)+68.05$$

(24)

where x_i (i = 1, 2…., 9) are input variables that are mathematically formulated using the weights and b1, as demonstrated in Table 4, and can be derived as shown in Equation (25) below.

$$\begin{array}{c}{x}_1=-5.2400tanh(0.9218B-0.2065S+2.0207D-2.1869L+0.9217T+4.2406PF+3.4291Q+0.0233DIS-2.1843RMR-1.7374)\\ x_2=-0.6904tanh(-0.1981B+1.2497S-1.1091D-3.1843L-0.1999T-1.4348PF-0.4071Q+0.1439DIS+2.1539RMR-3.5815)\\ x_3=3.3269tanh(2.9352B-0.5492S-2.0234D-0.2689L+4.7197T+2.0032PF+0.5673Q+2.1549DIS-1.7423RMR+0.5541)\\ x_4=-3.7788tanh(2.8495B-5.3477S-2.1881D+2.2159L+4.8837T+1.1803PF-5.0623Q+5.5159DIS-0.1145RMR+3.0744)\\ x_5=3.3778tanh(1.9090B-4.3624S-0.0267D+12.0182L+1.8981T-3.2526PF-4.5959Q+6.2092DIS-1.3156RMR+3.9849)\\ x_6=8.3641tanh(-1.3518B+1.6233S-0.2834D-2.7341L-1.3456T+0.1097PF+0.5537Q-3.7619DIS+0.3139RMR-3.8069)\\ x_7=4.0458tanh(1.7754B-1.8998S-0.0859D+3.5352L+1.8467T+0.4503PF-7.0705Q+2.9821DIS+0.5262RMR-1.5019)\\ x_8=-0.2037tanh(0.4929B+0.1933S-0.0307D-7.3747L+0.2709T-0.0737PF-6.3318Q+0.1578DIS-0.1736RMR-0.0216)\\ x_9=1.1364tanh(1.1422B+10.6302S-1.0916D-3.15188L+1.0554T+2.7472PF-0.9890Q+4.0753DIS+0.1246RMR+4.9379)\end{array}$$

(25)

For the MVO-ANN model, the generated equation is as presented in Equation (26).

$$AO{p}_{MVO-ANN}=16.55tanh\left({\displaystyle \sum _{i=1}^9{x}_i+1.3982}\right)+68.05$$

(26)

where y_i (i = 1, 2…., 9) are input variables that are mathematically formulated using the weights and b1, as highlighted in Table 4, and can be derived as shown in Equation (27).

$$\begin{array}{c}{y}_1=0.6818tanh(3.2593B+7.1832S-2.4079D-0.1395L+3.9862T-5.9387PF-0.9281Q+3.3369DIS+4.4127RMR+3.6026)\\ y_2=4.7120tanh(-1.6501B-1.1915S-0.1357D-0.1925L-0.9058T-0.1382PF-1.9574Q-1.0608DIS+0.8722RMR+2.7622)\\ y_3=-1.4915tanh(-1.0329B-0.6646S-0.2764D-3.3653L-1.1693T-2.7454PF-1.7059Q+2.2366DIS-0.7899RMR-1.8593)\\ y_4=-1.7079tanh(0.1325B-2.7229S+0.7861D+0.7534L-0.5904T+2.8602PF-2.0336Q-1.0599DIS-3.9818RMR+1.4799)\\ y_5=2.8986tanh(2.1845B-0.1519S+1.0670D-0.5641L+1.9426T+0.4679PF+0.9504Q+3.7224DIS-0.0898RMR-1.4936)\\ y_6=-1.6146tanh(0.8432B+2.5071S+0.8191D+0.3777L-0.9472T-2.9196PF-0.1334Q+4.8667DIS+6.8262RMR-1.4919)\\ y_7=-2.9749tanh(-0.0580B+3.7126S-0.1745D-3.3070L-0.8605T+0.9504PF+1.4986Q-3.7591DIS+0.7592RMR-2.5205)\\ y_8=0.8277tanh(-2.9660B-3.6424S+3.3059D+0.3393L-2.3650T+9.7145PF+1.8197Q+0.6470DIS+1.3214RMR+3.6159)\\ y_9=-4.7780tanh(1.008B-2.3431S+0.4299D+1.1599L+0.8472T-0.1035PF-1.2063Q+3.82278DIS-0.0846RMR+3.5629)\end{array}$$

(27)

When applying the proposed ANN-aided mathematical equations for predicting the AOp in construction rock excavation blasting, it is pertinent to note that the parameters x₁–x₉ and y₁–y₉ are in a normalized form and must be denormalized back to the original values after the calculation.

5.4. Analysis of the Relative Importance of Input Parameters on AOp Generation

To predict the AOp from construction rock excavation blasting, the significance of the influential parameters of the model to the AOp was investigated to avoid developing a complex model with little or no practical value. In this study, the cosine amplitude method (CAM) [49] was adopted to investigate the effects of the nine input parameters on AOp. The relevance importance between the parameters and the AOp is calculated as shown in Equation (28).

$$RI=\frac{{\displaystyle {\sum }_{i=1}^n(X_i{Y}_i)}}{\sqrt{{\displaystyle {\sum }_{i=1}^n{X}_i{}^2{\displaystyle {\sum }_{i=1}^n{Y}_i{}^2}}}}$$

(28)

where RI denotes the relative significance of each input parameter, X_i denotes the set of the nine input parameters, Y_i denotes the output of the model (AOp), and n is the total number of employed datasets.

Based on the calculated relevance importance scores for the parameters, the significance of each model input parameter on AOp generation from construction rock excavation blasting was appraised, as shown in Figure 10. From Figure 10, it can be concluded that all model input parameters have significant relevance to AOp. Therefore, these parameters must be considered when predicting the AOp at construction sites. It should also be noted that the rock mass rating (RMR) and powder factor (PF) have the most and least impact on the AOp, respectively. The two important parameters usually employed when planning a blasting round at construction sites (charge per delay (Q) and monitoring distance (DIS)) also have significant relevance to AOp, with RI values of 0.9797 and 0.9564, respectively.

5.5. Practical Engineering Validation of the Developed AOp Prediction Model Equations

To validate the developed ANN-based equations, five datasets were randomly selected from both the datasets used in the development of the models, and the AOp was predicted using the developed equations. The results were compared with those of the MLR, USBM empirical equations, and field measurements, as presented in

Table 5. Validation of the ANN-based equations for AOp prediction.

Data Point	Field Measurement	SMA-ANN (Equation (24))	MVO-ANN (Equation (26))	MLR (Equation (15))	USBM (Equation (17))
1	59.5	60.4	63.1	64.2	68.3
2	58.5	58.6	58.9	63.3	61.2
3	77.4	77.0	78.9	70.6	71.9
4	67.6	67.3	63.4	66.1	69.4
5	73.9	73.7	74.7	73.3	72.6
6	84.6	84.6	84.6	56.2	59.0
7	60.2	59.8	61.0	67.2	64.6
8	58.8	57.1	56.7	66.1	63.8
9	71.5	71.0	71.4	71.9	70.9
10	78.9	78.7	78.0	67.5	71.6

. In Table 5, we can observe that the developed ANN-based equation predictions are the closest to the field measurements; specifically, the optimum SMA-ANN model equation predicted the AOps with the highest accuracy. Nevertheless, both the MLR and USBM equation predictions were close to field measurements at data points 5 and 9.

However, similar to other developed models, the application of the proposed ANN-based SMA-ANN and MVO-ANN equations could be limited by variations in the datasets employed for model generation in this study. The findings in Table 5 are supported by a graphical representation of the validation results of the different models developed against the field measurements, as shown in Figure 11. The AOp values estimated by the optimum SMA-ANN appeared to be closest to the field measurements at all data points.

6. Conclusions

Researchers have made efforts to develop different models aimed at forecasting unpleasant environmental problems caused by blasting, such as blast-induced ground vibration, air overpressure, and flyrock occurrence. However, most of their efforts have been directed towards mine production blasting, while the unwanted environmental problems from rock excavation by blasting at construction sites have rarely been studied. In addition, despite the large number of soft computing models for the prediction of airblast overpressure in the literature, most of these models are not in the form of explicit and closed-form equations and are limited to one or two sites at most. This study proposed closed-form and explicit ANN-based soft computing equations for forecasting AOp generation in multiple-construction rock excavation blasting with high accuracy. The proposed model used ANN as its base model and improved its performance by using two metaheuristic algorithms, SMA and MVO. The predictive performances of SMA-ANN and MVO-ANN were evaluated using six error metrics. To evaluate the robustness of the developed ANN-based hybrid models, their performance was compared with those of the stand-alone ANN, MLR, and USBM models. Closed-form and explicit equations were generated using the bias and weights of the training network to ensure the general applicability of the developed ANN-based model. The following conclusions can be drawn from the findings of this study:

(1)

The SMA-ANN model had the best performance among all the developed models, with MAE, MSE, RMSE, MAPE, RMSE, and R² values of 2.363, 10.803, 3.286, 3.576, 0.051, and 0.809 for the training datasets and 2.102, 8.084, 2.843, 3.245, 0.045, and 0.867 for the test datasets, respectively. This is followed closely by the MVO-ANN and ANN models.

(2)

The performance of the two linear models (MLR and USBM) was poor, as they both underestimated or overestimated the AOp values at most of the data points. Hence, they are not recommended for use in forecasting AOp at construction sites.

(3)

Based on the relevance importance scores, both controllable and uncontrollable factors (RMR) are important for predicting AOp at construction sites.

(4)

The developed ANN-based closed-form and explicit equations proved to be reliable and recommended for use in forecasting AOp at construction sites, provided the range of data is the same or similar to that used in this study.

(5)

Although the developed ANN-based models’ accuracy is ideal for this study, it would be significant if meteorological factors such as wind speed and direction, as well as temperature, were considered. Future studies may benefit from incorporating drilling, blasting, explosives, geological, and meteorological factors for more robust models.