A Solution Surface in Nine-Dimensional Space to Optimise Ground Vibration Effects Through Artificial Intelligence in Open-Pit Mine Blasting

Abstract

In this study, we model a solution surface, with each point having nine components using artificial intelligence (AI), to optimise the effects of ground vibration during blasting operations in an open-pit diamond mine. This model has eight input parameters that can be adjusted by blasting engineers to arrive at a desired output value of ground vibration. It is built using the best performing artificial neural network architecture that best fits the blasting data from 100 blasting events provided by the Debswana diamond mine. Other AI algorithms used to compare the model’s performance were the k-nearest neighbour, support vector machine, and random forest—together with more traditional statistical approaches, i.e., multivariate and regression analyses. The input parameters were burden, spacing, stemming length, hole depth, hole diameter, distance from the blast face to the monitoring point, maximum charge per delay, and powder factor. The optimised model allows for variations in the input values, given the constraints, such that the output ground vibration will be within the minimum acceptable value. Through unconstrained optimisation, the minimum value of ground vibration is around 0.1 mm/s, which is within the vibration range caused by a passing vehicle.

1. Introduction

Blasting is commonly used in mining, quarrying, and construction to break apart hard rocks and enable easier excavation and removal. Only 20 to 30 percent of the explosive energy during a blast is used for rock breaking, with the remainder contributing to negative environmental impacts, such as ground vibration, airblast, and flyrock [1,2]. The vibrations induced by blasting affect neighbouring residents and may damage structures [3]. Consequently, predicting blast-induced ground vibrations becomes imperative to manage and mitigate the undesirable effects of blasting. Ground vibrations originating from a blast are quantified in terms of peak particle velocity (PPV), measured in mm/s, at a designated ground position. Influencing factors for ground vibrations are categorised into controllable and uncontrollable parameters. The former pertains to human-made elements, such as explosive properties and blast design parameters, while the latter involves natural occurrences, such as geological conditions and rock-mass properties [4,5].

We present a method to establish the relationship of ground vibration versus input parameters as shown below

$$GV=f(S,B,T,L,D,MC,Pf,Di)$$

(1)

where $GV$ = ground vibration, S = spacing, B = burden, T = stemming, L = hole depth, D = hole diameter, $MC$ = maximum charge per delay, $Pf$ = powder factor, and $Di$ = distance from the blast point to the monitoring point.

Equation (1) above has been the subject of many studies aiming to define the function $f\left(\right)$, that is, to establish the effects of input parameters versus the output parameter in a blasting event in the mines. Most mining companies in the world use different software in estimating the function $f\left(\right)$, namely, Blast Logic, Mining Excellence, O-Pit Blast, Surpac, etc. These can help them in their blast design efforts, that is, provide values of the input parameters and allow them to estimate the outputs of blasting, e.g., rock fragmentation, ground vibration, airblast, and flyrock. Most of these software use empirical methods in defining the function $f\left(\right)$.

Our solution is to use machine learning to model the function $f\left(\right)$, and we are not the only ones who attempted to use this approach. What is unique about our approach is that we provide a solution surface, which is the problem’s solution space, to further define the function $f\left(\right)$. In previous studies, researchers have only defined the function $f\left(\right)$ as a black box that is capable of predicting the output parameter given the input parameters, without providing a solution space. Our solution space enables us to perform prediction as well as two other things as follows:

1.

Perform optimisation computations to find the minimum ground vibration using random initial points.

2.

Perform blast design by setting the desired value of ground vibration and searching in the solution space for the values of the corresponding input parameters. Alternatively, we can assign the value of ground vibration and some input parameters as constraints, and we can search in the solution space for the values of the remaining input parameters.

To the best of our knowledge, no other previous studies could accomplish this.

1.1. Empirical Methods

Numerous empirical methodologies for predicting ground vibration due to blasting have been developed by researchers [6,7,8]. These techniques use data obtained from different blasting operations, in certain ground conditions which cannot be generalised for different rock types. There are two major shortfalls related to empirical methods. First, they have a limited number of inputs, normally only two—the maximum charge per delay and distance from the blast point to the monitoring point. Second, they have very low accuracy. Artificial intelligence methods provide generalisability, especially in terms of the number of inputs to the model. However, in terms of accuracy, the results may vary depending on the algorithm used. Moreover, machine learning methods generally provide much higher accuracy compared to empirical methods. From our study, accuracy comparisons are stated as follows. The United States Bureau of Mines (USBM) empirical model has an accuracy of 12.6%, the accuracy of our artificial intelligence model ranges from 54.6% to 93.6%, and the statistical method has an accuracy of 66%.

Adding more input parameters in modelling ground vibration using empirical methods can be very difficult to achieve with high accuracy because of the complex, non-linear relationship between the input and output parameters. This could be the reason why empirical methods typically account for only two parameters and fail to incorporate the full range of factors affecting ground vibration. New approaches to estimating ground vibrations from blasting include artificial intelligence systems that consider many parameters influencing the outcomes of blasting [9,10,11,12].

1.2. Machine Learning Methods

Lawal et al. [13] utilised an ordinary artificial neural network (ANN), particle swarm optimisation (PSO), and dragonfly-optimised ANN to forecast how tunnel blasting may affect the environment. The PSO-ANN model, with an architecture of 4-15-15-1, yielded the best performance, exhibiting a coefficient of determination ($R^2$) of 1, a mean square error (MSE) of 0.125, and a variance accounted for (VAF) of 98.29. In another study, Lawal et al. [14] used blast data from five granite quarries to predict ground vibration by employing gene expression programming (GEP), a sine cosine algorithm-optimised artificial neural network (SCA-ANN), and adaptive neuro-fuzzy inference system (ANFIS) models, using a total of 100 datasets for model development. The obtained $R^2$ values for the SCA-ANN, GEP, and ANFIS models were 0.999, 0.989, and 0.997, respectively.

Zhang et al. [15] applied five machine learning classifiers for predicting ground vibration, namely, the chi-squared automatic interaction detection (CHAID), ANN, support vector machine (SVM), random forest (RF), and classification regression tree (CART), finding RF demonstrates substantially superior performance over the other regression models investigated. Qiu et al. [16] applied an extreme gradient boosting (XGBoost) model to predict blast-induced ground vibration using 150 data sets with thirteen input parameters. The grey wolf optimisation (GWO), whale optimisation algorithm (WOA), and Bayesian optimisation algorithm (BO) were used to fine-tune the hyper-parameters of the XGBoost model. Their investigation revealed that the suggested WOA-XGBoost model performed better in comparison to other machine learning (ML) models.

Nguyen et al. [17] utilised a novel hybrid model combining the hierarchical k-means clustering algorithm with an ANN (HKM-ANN). The model was compared to fuzzy c-means clustering support vector regression (FCM-SVR), finding that the HKM-ANN model performed better than the FCM-SVR model in forecasting ground vibration. Yang et al. [18] employed ANFIS optimised by PSO and a genetic algorithm (GA) to forecast ground vibration resulting from blasting, utilising data from two Iranian quarry mines. Their model demonstrated an accuracy of 0.979, a root mean square error ($RMSE$) of 0.240, and a mean absolute error (MAE) of 0.199 in predicting ground vibration. Other researchers have employed hybrid artificial intelligence methodologies, predominantly ANNs, to model blast-induced ground vibrations and other blast-induced impacts and rock engineering problems [19,20,21,22,23,24,25,26,27,28].

This study uses an ANN model as a solution surface in a nine-dimensional space to minimise the effects of ground vibration during blasting operations in an open-pit diamond mine. This solution surface becomes the solution space such that each point on the surface is defined by nine components, i.e., eight input parameters and one output parameter ground vibration. An ANN model is designed for predicting ground vibrations during open-pit mine blasting, using data derived from the Debswana Orapa Diamond Mine in Botswana. The ANN model was chosen as the best-performing model among three other machine learning methods. The model has eight input parameters, which include both blast design parameters and explosive properties. The model’s performance is compared against other machine learning algorithms and multivariate regression analysis. Sensitivity analysis is also conducted to determine the relative influence of the input parameters on the output.

We explicitly list our contributions in this paper as follows:

• We present data from Debswana Diamond Company recorded from 100 blast events.
• We develop machine learning models, each with five different architectures, eight input parameters, and one output of ground vibration. The four models are compared against a statistical method.
• We optimise the architecture of the best performing machine learning model using the Monte Carlo method.
• A solution space is created from the optimised machine learning model. Other machine learning models did not yield a solution space, as their results only predicted the output parameter ground vibration. Our solution space is capable of inverse solution, i.e., we can search for the solution space, given the input parameters and the expected output, to help blast design engineers in adjusting the input parameters to arrive at an expected output of ground vibration.
• We optimise ground vibration using the gradient descent method from the created solution space.
• Sensitivity analysis is performed using a statistical method and the results are confirmed from the created solution space.

The machine learning model allows us to predict the output ground vibration from the given eight input parameters by locating a unique point in the solution space. Because of this model, one can also predict not just the output parameter, but also an input parameter when seven other input parameters are given together with their corresponding output ground vibration. In this way, this model can be very useful to blast design engineers by setting a target output ground vibration and searching for the corresponding input parameters in the solution space. This is a kind of inverse solution approach and can be very useful to blast design engineers.

1.3. Mine Case Study

The Orapa Diamond Mine in Botswana was discovered in 1967 by De Beers. The kimberlite igneous rocks span an area of 1.18 square kilometres, with two merging diatremes, each with preserved crater sedimentary facies known as the southern and northern lobes. These lobes overlay Archaean granite and belong to the Karoo Supergroup, incorporating Stormberg-aged basalt lava and tracing back approximately 93 million years. The anticipated yield of the Orapa mine is around 153 million carats from a total of 205 million tonnes of mineral reserves. Figure 1 depicts the current layout of the Orapa Diamond Mine.

The mine utilises a staggered blasting pattern. The predominant explosives used are S135B emulsion and pentolite boosters.The mine has a bench height of 15 m and makes use of blast holes with diametres of 127, 165, and 250 millimetres. Each blast comprises between 15 and 25 rows, with each row containing between 40 and 60 holes. The loading of the fragmented materials is carried out using shovels and hydraulic excavators, while dump trucks are used for hauling.

2. Materials and Methods

This section explores the materials and methodologies employed in this study, focusing on the datasets collected from the mining operation, the machine learning techniques applied, and the optimisation processes implemented.

2.1. Materials

A dataset with 100 blasting events was collected from the blasting operations of the Debswana Orapa Diamond Mine. The dataset consists of eight inputs, including the blast design parameters and explosive properties, and the ground vibration as the output, as shown in

Table 1. Description of the input and output parameters for the models.

Parameter	Type	Unit	Symbol	Min	Max
Burden	Input	m	B	4	6
Spacing	Input	m	S	5	8
Stemming length	Input	m	T	4	6
Hole depth	Input	m	L	12	15
Hole diameter	Input	mm	D	165	250
Distance from blast to monitoring point	Input	m	DI	438	1500
Maximum charge per delay	Input	kg	MC	216	552.6
Powder factor	Input	kg/m3	Pf	0.3	1.17
Ground vibration	Output	mm/s	GV	0.163	6.5

.

Ground vibration data in this study were recorded using seismographs placed at varying distances from the blast sources. These devices measured vibrations in three orthogonal components—radial, transverse, and vertical. The radial component corresponds to the direction of propagation of the vibration wave from the blast source, while the transverse and vertical components provide a complete three-dimensional representation of the vibration effects. This approach ensures that the dataset captures the full spectrum of vibrational impacts resulting from blasting activities. “Starting from 1984 to the present, Orapa had an average of around two blast events per month. The data used in the study span the period from 2018 to 2024, covering all blasting conditions and potential biases in this six-year period. There were a total of 144 blast events. However, some events have incomplete information and need to be discarded. This results into 100 events with complete information that are included in our dataset. In terms of the completeness and representativeness of the data collected, we present the range of values for each of the following parameters included in the dataset, as shown on Table 1: (a) distance from blast point to monitoring stations (ranging from 438 m to 1500 m), (b) hole depths (12 m to 15 m), and (c) hole diameter (127 mm, 165 mm, and 250 mm). These parameters are consistent with the range of values the Orapa mine used throughout its blast operations in the stated six-year period. Thus, this ensures the relevance and representativeness of the dataset used within the operational context”.

2.2. Methods

Data preprocessing involved cleaning and normalising the data to prepare it for analysis; the data was divided into training (80%), validation (10%), and testing (10%) sets. The machine learning models employed are the random forest, support vector machine, k-nearest neighbour (k-NN), and artificial neural network models. Additionally, the statistical model multivariate regression analysis (MVRA) was utilised. These models were built using Python 3.9.16 Specifically, scikit-learn libraries were used to model the SVM, k-NN, and RF models, and Keras library was used in Tensor Flow for modelling using an ANN. The gradient descent method, coupled with the Monte Carlo method, was used for the optimisation of ground vibration based on the best-performing ANN model. The cosine amplitude approach was applied for sensitivity analysis to determine the relative impact of input parameters on the output. The metrics used for evaluating the performance of all predictive models are the $RMSE$ and $R^2$. The $RMSE$ is calculated using the following formula:

$$RMSE=\sqrt{{\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N}{(y-y^{\prime })}^2}$$

(2)

The $\left(R^2\right)$ is determined by the following:

$$R^2={\left[{\displaystyle \frac{{\sum }_{i=1}^N(y-\overline{y})(y^{\prime }-{\overline{y}}^{\prime })}{{\sum }_{i=1}^N{(y-\overline{y})}^2{\sum }_{i=1}^N{(y^{\prime }-{\overline{y}}^{\prime })}^2}}\right]}^2$$

(3)

In these equations, y and $y^{\prime }$ represent the measured and forecasted values, respectively, while $\overline{y}$ and ${\overline{y}}^{\prime }$ denote the mean values.

For composite score calculation, $R^2$ is normalised. We note that higher values of $R^2$ indicate better performance. For the $RMSE$, lower values indicate better performance. To normalise the $RMSE$ values, we used the following:

$$\mathrm{Normalized}\mathrm{RMSE}=1-{\displaystyle \frac{\mathrm{RMSE}}{\mathrm{Max}\mathrm{RMSE}}}$$

(4)

Once both metrics are normalised, we compute a composite score by adding the normalised $R^2$ and normalised $RMSE$.

The Monte Carlo technique was employed to randomly navigate the solution space and find the best ANN configuration. This optimal ANN structure comprises 34 neurons in the first hidden layer, 15 in the second layer, and 50 in the third hidden layer. Upon establishing the optimal ANN model, we then use the gradient descent approach to find the minimum ground vibration and its corresponding input parameters. Gradient descent is a commonly employed iterative optimisation technique with applications spanning various computational fields [29]. The mathematical equation of the gradient descent method is shown by (5).

$$\begin{array}{cc}\hfill p_i^{\mathrm{new}}& =p_i\hfill \\ \hfill & -\eta \left({\displaystyle \frac{\mathrm{nnet}(p_1,\dots ,p_i,\dots ,p_N)-\mathrm{nnet}(p_1,\dots ,p_i^{\mathrm{old}},\dots ,p_N)}{p_i-p_i^{\mathrm{old}}}}\right)\hfill \end{array}$$

(5)

In this equation, $p_i^{\mathrm{new}}$ represents the updated parameter, while $p_i$ is the current parameter value being updated. The term $p_i^{\mathrm{old}}$ denotes the parameter value from the previous iteration. The learning rate is represented by $\eta$. The function $\mathrm{nnet}(·)$ represents the neural network model.

3. Results and Discussion

Table 2. Calculated performance indices for all the algorithms.

Method	${\mathit{R}}^2$	RMSE	Composite Score
k-nearest neighbor
n-neighbors = 10	0.728	1.030	1.454
n-neighbors = 20	0.643	1.180	1.329
n-neighbors = 30	0.528	1.670	1.084
n-neighbors = 40	0.550	1.884	1.049
n-neighbors = 50	0.582	2.002	1.050
Support vector machine
sigma = 1	0.675	1.130	1.374
sigma = 3	0.696	1.145	1.391
sigma = 5	0.720	1.035	1.445
sigma = 7	0.596	1.215	1.275
sigma = 9	0.590	1.245	1.260
Random forest
n-estimators = 5	0.904	0.510	1.768
n-estimators = 15	0.892	0.545	1.748
n-estimators = 25	0.865	0.605	1.705
n-estimators = 35	0.856	0.680	1.676
n-estimators = 45	0.848	0.760	1.646
Artificial neural network
model 1 (10 neurons)	0.941	0.286	1.864
model 2 (20 neurons)	0.912	0.315	1.829
model 3 (30 neurons)	0.898	0.456	1.778
model 4 (40 neurons)	0.890	0.410	1.800
model 5 (50 neurons)	0.862	0.472	1.737
Multivariate regression analysis	0.664	3.760	0.664

shows the performance indices of the models. The ANN model with an 8-10-1 architecture showed the best performance, with the lowest $RMSE$ (0.286), the highest accuracy $R^2$ (0.941), and the highest composite score (1.864), establishing it as the best model. The order of performance of the models according to their composite score is ANN (1.864), RF (1.768), k-NN (1.454), SVM (1.445), and MVRA with the lowest composite score of 0.664.

Figure 2 shows the variation in the $RMSE$ for different models with their hyper-parameters. Figure 2a illustrates the variation in the $RMSE$ with the number of neighbours for the k-NN model. As the number of neighbours decreases, the $RMSE$ also decreases, with the optimum performance found at k = 10 (lowest $RMSE$). For the RF model, increasing the number of trees results in a higher $RMSE$. The optimum model was found at 10 trees (lowest $RMSE$), as shown in Figure 2b. In Figure 2c, the $RMSE$ decreases as the number of neurons decreases, leading to an optimum ANN model with 10 neurons (lowest $RMSE$). For the SVM model, smaller sigma values lead to an increased $RMSE$, which is pattern consistent with larger sigma values as well. Medium sigma values result in a lower $RMSE$, with the optimum model being the SVM polynomial kernel, with $\sigma$ = 5.

Figure 3a–e depicts the correlation between the predicted and measured ground vibrations. The solid line represents a 1:1 slope, and it can be observed that the data points for the ANN model cluster closely around this line, indicating superior prediction capability compared to other models. Furthermore, a comparison of the predicted and measured ground vibration for each algorithm is shown in Figure 4. The graph illustrates that the predictions generated by the ANN closely align with the measured values, indicating the ANN’s proficiency in predicting these parameters compared to other models.

The ANN’s ability to capture more complex, non-linear, and high-dimensional relationships arises from its multi-layer architecture with several neurons per layer, enabling it to approximate intricate functions within high-dimensional solution spaces. This is compared to RF and SVM, which also handle non-linearity but rely on ensemble averaging or kernel tricks, respectively. During training, the ANN continuously adjusts multiple weights, which can reach up to thousands or even millions, depending on the complexity of the solution. This is where the computational power of ANN lies; the number of adjustable weights is parallel to the degrees of freedom of the designed solution. Our chosen ANN architecture (8-10-1) demonstrated the best generalisation on unseen data, achieving an R² of 0.941, due to its capacity to form flexible decision boundaries and learn hierarchical feature representations that align closely with the blast-induced ground vibration patterns. The RF, while also capable of handling non-linearities through ensemble learning, is slightly less effective than ANN in capturing very complex interactions. The k-NN, which relies on the proximity of data points, performs adequately but struggles with high-dimensional data. The SVM, though powerful in high-dimensional spaces and capable of non-linear modelling using kernel functions, often requires the meticulous tuning of hyper-parameters to perform optimally. Lastly, the MVRA, being a traditional statistical approach, is limited by its linear assumption and cannot adequately capture the non-linear relationships present in the data, resulting in the poorest performance among the models.

The best-performing model in Table 3 is the PSO-ANN by Lawal et al. [13], which achieved an $R^2$ of 1.00 using 56 datasets with four inputs, namely, charge per delay (Q), number of holes (N), DI, and rock mass rating (RMR). This model’s superior performance can be attributed to the optimisation capabilities of the PSO technique combined with the ANN, which likely captures non-linearities more effectively than other models. In contrast, the lowest-performing model is the ANN by Dindarloo [30], with an $R^2$ of 0.81 using only 15 datasets and five inputs (D, L, N, B, and S). Its limited dataset size and potentially less complex model architecture might have restricted the model’s ability to generalise and capture variability in the data, leading to lower predictive accuracy. In the middle range, the HKM-ANN by Nguyen et al. [17] achieved an $R^2$ of 0.98 using 185 datasets with six inputs (S, PF, B, DI, and maximum charge per delay (MC)). This model combines hierarchical k-means clustering with an ANN, enabling it to handle larger and more diverse datasets effectively. Its performance is close to the best-performing models, indicating that hybrid techniques can significantly enhance predictive capabilities.

Comparing these with our results, our ANN model achieved an $R^2$ of 0.941 with 100 datasets and eight inputs. It performs well and aligns closely with other high-performing models, such as the ANN by Armaghani et al. [31], which achieved an $R^2$ of 0.97 with 109 blasting events. This suggests that our model may attain comparable accuracy with minor adjustments, such as including more input parameters and increasing the dataset size. However, hybrid models that use advanced techniques and larger datasets, including PSO-ANN ($R^2$ = 1.0, Lawal et al. [13]) and HKM-ANN ($R^2$ = 0.98, Nguyen et al. [17]), perform slightly better than our model. These results imply that the accuracy and generalisability of our model can be improved by investigating hybrid models and adding more inputs, allowing it to capture more complex interactions and improve generalisability, as seen in the best-performing models.

Table 3 highlights the significant impact of dataset size, the number of input parameters, and the type of model used on the performance of the predictive models. Models using hybrid techniques and larger datasets tend to perform better, as they can capture complex patterns and interactions within the data more effectively.

Table 3. Performance indices from other studies.

Best Model	Other Models	Inputs	Dataset	${\mathit{R}}^2$	Reference
ANFIS		L, B, S, T, Q, DI	25	0.89	[25]
RF	CART, CHAID,	BS, DI, T	102	0.94	[15]
	ANN, SVM, RF	MC, PF, L
HKM-ANN	ANN, SVR, FCM-ANN	S, Pf, B	185	0.98	[17]
	HKM-SVR, FCM-SVR	DI, MC
ANFIS-GA	ANFIS, ANFIS-PSO	B, S, T	86	0.98	[18]
	USBM, Indian Standard	Pf, MC, DI
GEP		BS, L, T, Pf, Q, DI	102	0.88	[22]
SVM	USBM, MLR, PSO power, PSO linear	DI, MC	80	0.96	[32]
WOA-XGBoost	GWO-XGBoost, BO-XGBoost,	D, L, B, S, MC	150	0.97	[16]
	CatBoost, RF, GBR	CL, DI, BI
		E, PR, Pv, VOD, DOE
ANFIS	ANN	DI, MC	109	0.97	[31]
PSO-ANN	ANN, DA-ANN	Q, Nh, DI, RMR	56	1.00	[13]
ANN		MC, DI, TC	20	0.93	[33]
ANN	MVRA, Indian Standard	L, MC	174	0.99	[34]
	Langefors-Kihlstrom, USBM	B, S
	General Predictor, Ambraseys-Hendron	DI, BI, E, PR, Pv
GRNN	USBM, CMRI, Indian Standard	MC, DI	14	0.99	[35]
	Langefors-Kihlstrom, Ambraseys-Hendron
ANN	USBM, Indian Standard, MVRA	MC, DI, BS, L, T, D, Pf	180	0.99	[36]
	Langefors-Kihlstrom, Ambraseys-Hendron
FIS	USBM, Indian Standard	B, S	120	0.94	[37]
	Langefors-Kihlstrom, Ambraseys-Hendron	T, N
	CMRI, Ghosh-Daemen 1, Ghosh-Daemen 2	MC, DI
	MVRA, General Predictor
ANN	GEP	D, L, N, B, S	15	0.81	[30]
		RDI, HDI, T, Q
SVM	USBM, Indian Standard, MVRA	DI, MC	174	0.96	[38]
	Langefors-Kihlstrom, Ambraseys-Hendron
	Ghosh-Daemen, CMRI, General Predictor
FS-RF	FS-BN, Langefors-Kihlstrom	MC, DI, B, D, TC, S	102	0.90	[5]
	Ghosh-Daemen, Roy, Indian Standard	L, T, Sd, N, Pf, Q
ANN-KNN	ANN, USBM	MC, DI	75	0.88	[23]
RVR-GWO	BA-GWO	MC, BS, D, T	95	0.84	[28]

USBM, United States Bureau of Mines; MLR, Multi-Linear Regression; GBR, Gradient Boosting Regression; GRNN, Generalised Regression Neural Network; GEP, Gene Expression Programming; CMRI, Central Mining Research Institute; FIS, Fuzzy Inference System; FS, Feature Selection; RVR, Relevance Vector Regression; GWO, Grey Wolf Optimisation; BA, Bat-inspired Algorithm; BN, Bayesian Network; BS, Burden to Spacing Ratio; Q, Charge per Delay; Sd, Subdrill; TC, Total Charge; DI, Distance; N, Number of holes; MC, Maximum Charge per Delay; BI, Blastability Index; E, Young’s Modulus; PR, Poisson’s Ratio; VOD, Velocity of Detonation; DOE, Density of Explosives; Pv, P-wave Velocity.

Table 3. Performance indices from other studies.

Best Model	Other Models	Inputs	Dataset	${\mathit{R}}^2$	Reference
ANFIS		L, B, S, T, Q, DI	25	0.89	[25]
RF	CART, CHAID,	BS, DI, T	102	0.94	[15]
	ANN, SVM, RF	MC, PF, L
HKM-ANN	ANN, SVR, FCM-ANN	S, Pf, B	185	0.98	[17]
	HKM-SVR, FCM-SVR	DI, MC
ANFIS-GA	ANFIS, ANFIS-PSO	B, S, T	86	0.98	[18]
	USBM, Indian Standard	Pf, MC, DI
GEP		BS, L, T, Pf, Q, DI	102	0.88	[22]
SVM	USBM, MLR, PSO power, PSO linear	DI, MC	80	0.96	[32]
WOA-XGBoost	GWO-XGBoost, BO-XGBoost,	D, L, B, S, MC	150	0.97	[16]
	CatBoost, RF, GBR	CL, DI, BI
		E, PR, Pv, VOD, DOE
ANFIS	ANN	DI, MC	109	0.97	[31]
PSO-ANN	ANN, DA-ANN	Q, Nh, DI, RMR	56	1.00	[13]
ANN		MC, DI, TC	20	0.93	[33]
ANN	MVRA, Indian Standard	L, MC	174	0.99	[34]
	Langefors-Kihlstrom, USBM	B, S
	General Predictor, Ambraseys-Hendron	DI, BI, E, PR, Pv
GRNN	USBM, CMRI, Indian Standard	MC, DI	14	0.99	[35]
	Langefors-Kihlstrom, Ambraseys-Hendron
ANN	USBM, Indian Standard, MVRA	MC, DI, BS, L, T, D, Pf	180	0.99	[36]
	Langefors-Kihlstrom, Ambraseys-Hendron
FIS	USBM, Indian Standard	B, S	120	0.94	[37]
	Langefors-Kihlstrom, Ambraseys-Hendron	T, N
	CMRI, Ghosh-Daemen 1, Ghosh-Daemen 2	MC, DI
	MVRA, General Predictor
ANN	GEP	D, L, N, B, S	15	0.81	[30]
		RDI, HDI, T, Q
SVM	USBM, Indian Standard, MVRA	DI, MC	174	0.96	[38]
	Langefors-Kihlstrom, Ambraseys-Hendron
	Ghosh-Daemen, CMRI, General Predictor
FS-RF	FS-BN, Langefors-Kihlstrom	MC, DI, B, D, TC, S	102	0.90	[5]
	Ghosh-Daemen, Roy, Indian Standard	L, T, Sd, N, Pf, Q
ANN-KNN	ANN, USBM	MC, DI	75	0.88	[23]
RVR-GWO	BA-GWO	MC, BS, D, T	95	0.84	[28]

USBM, United States Bureau of Mines; MLR, Multi-Linear Regression; GBR, Gradient Boosting Regression; GRNN, Generalised Regression Neural Network; GEP, Gene Expression Programming; CMRI, Central Mining Research Institute; FIS, Fuzzy Inference System; FS, Feature Selection; RVR, Relevance Vector Regression; GWO, Grey Wolf Optimisation; BA, Bat-inspired Algorithm; BN, Bayesian Network; BS, Burden to Spacing Ratio; Q, Charge per Delay; Sd, Subdrill; TC, Total Charge; DI, Distance; N, Number of holes; MC, Maximum Charge per Delay; BI, Blastability Index; E, Young’s Modulus; PR, Poisson’s Ratio; VOD, Velocity of Detonation; DOE, Density of Explosives; Pv, P-wave Velocity.

3.1. Sensitivity Analysis

Sensitivity analysis was used to determine the relative influence of the input parameters on the output using the cosine amplitude method [39]. The strength of the relation between the dataset $x_i$ and $xj$ is given by the following:

$$r_{ij}={\displaystyle \frac{{\sum }_{k=1}^m{X}_{ik}{X}_{jk}}{\sqrt{{\sum }_{k=1}^m{X}_{ik}^2{\sum }_{k=1}^m{X}_{jk}^2}}}$$

(6)

where $x_i$ and $xj$ are the inputs and outputs, respectively, and m is the dataset number.

In Figure 5, it can be inferred that spacing is the most influential input parameter on ground vibration. It is noted that the hole diameter is the least influential parameter on ground vibration. This observation is consistent with findings from other studies, which have also demonstrated that spacing exerts a greater influence on ground vibration than hole diameter [5,16,36]. The high sensitivity of spacing suggests that even minor adjustments can lead to significant changes in vibration levels, providing a powerful lever for engineers to control blast outcomes. On the other hand, the relatively low sensitivity of hole diameter indicates that it can be considered a secondary factor, allowing engineers to focus their optimisation efforts on more influential parameters.

The second most influential parameter is maximum charge per delay, followed by stemming, burden, and powder factor. The first four most influential parameters have the greatest effect compared to the rest of the parameters, and therefore, they must be carefully considered in the blast design when ground vibration is a desired output to be optimised. Furthermore, within this group itself, the order of parameter influence must be considered to effectively achieve the desired ground vibration optimisation. The same method of sensitivity consideration towards the influence of ground vibration output can be applied to the four least influential parameters. The sensitivity analysis approach in this section is validated in the sensitivity analysis of the solution space in Section 3.2 “Analysis of the Optimisation Results”.

3.2. Analysis of the Optimisation Results

Figure 6a through Figure 7a display the results of the optimisation procedure, starting with seven initial points (depicted as red dots) chosen at random and converging towards the optimised value (illustrated as a blue dot), which is the lowest point in the surface. The optimised blast design parameters are as follows: powder factor (0.58), maximum charge per delay (454.09), burden (8.22), spacing (8.30), stemming (7.16), distance from blast point to monitoring point (2399), hole depth (18.25), and diameter (183). Therefore, the lowest point on the solution surface corresponds to the minimum value of ground vibration predicted by the model (approximately 0.1 mm/s). Every point in the solution surface has nine components, i.e., eight components of the eight input parameters plus the output component of the ground vibration.

The solution surface generated from the optimal ANN model (with 34 neurons in the first hidden layer, 15 in the second layer, and 50 in the third hidden layer) using gradient descent was used to predict ground vibration. The $R^2$ and $RMSE$ values are 0.96 and 0.21, respectively, showing high accuracy and low error.

Table 4. Measured and predicted ground vibration values using the optimal ANN model.

Measured Ground Vibration	Predicted Ground Vibration
0.312	0.480
1.990	1.880
3.195	3.077
3.551	3.576
0.145	0.180
0.388	0.361
0.511	0.491
3.915	4.076
5.226	5.063

shows the values for the measured and predicted ground vibration from the optimal ANN model.

From the sensitivity analysis, spacing is the most influential parameter. Because of this, we compare spacing with all other seven input parameters kept constant to observe the variations in the values of the output parameter in a two-dimensional plot. We assign spacing to be the x-axis, while ground vibration is assigned as the y-axis.

Figure 7a shows that as spacing decreases with stemming length fixed at 7.16, ground vibration initially fluctuates and finally decreases towards an optimal spacing value for minimal vibration, after which it increases again. The solution space also confirms the results of the sensitivity analysis. As can be seen in Figure 7b, generated by varying spacing, fixing all other parameters, and taking a slice through the lowest, optimised point in the solution space, the maximum charge per delay, stemming, burden, and powder factor are the most influential input parameters after spacing, and they have the most fluctuations in Figure 7b. This indicates high sensitivity, as changes in these parameters significantly impact ground vibration. Hole diameter is the least influential parameter, as showing the least variations suggests lower sensitivity.

The significance of these results lies in their practical application for blasting engineers seeking to minimise the environmental and structural impacts of ground vibrations during blasting operations. By providing a set of optimised parameters, the model enables engineers to design blasts that adhere to safety regulations and reduce the risk of damage to nearby structures and communities. Reducing ground vibrations not only mitigates the risk of structural damage but also enhances the efficiency of the blasting process. Minimising ground vibration may lead to a greater part of the total energy in blasting to be concentrated more towards its main goal, that is, rock fragmentation.

4. Conclusions

This study successfully demonstrates the application of machine learning techniques to predict and optimise ground vibrations induced by blasting operations at the Debswana Orapa diamond open-pit mine. Among the evaluated models—ANN, k-NN, SVM, RF, and the MVRA—the ANN model emerged as the most effective, evidenced by its highest predictive accuracy, with an $R^2$ value of 0.941 and a minimal $RMSE$ of 0.286. Using the ANN algorithm, we optimise the architecture of the ANN model using the Monte Carlo method, resulting in an architecture consisting of three hidden layers with 34, 15, and 50 neurons, respectively. We then optimise the resulting ANN model using the gradient descent method to find the minimum ground vibration, which is 0.1 mm/s. The solution space is therefore capable of predicting ground vibration given the input parameters, as well as setting the desired output value of ground vibration and looking for the corresponding values of the input parameters. Thus, the solution space demonstrates its efficacy in minimising, predicting, and designing ground vibrations. Sensitivity analysis confirmed the results provided by the solution space. Although the findings of this study are specific to the Orapa Diamond Mine, the methodologies employed are designed to be scalable and adaptable to other mining operations. To model other mining sites, we will use the results of this study as the baseline model. Then, we will incorporate site-specific data to correct the existing model and iteratively validate the results until the error becomes minimal. When this is achieved, the newly modified model would apply to the specific mining site.