1 Introduction

The Earth is richly endowed with mineral reserves (raw materials) which are beneficial to the existence of mankind. These raw materials include precious metals such as gold, diamond, silver, bauxite, iron, nickel, manganese, cobalt, platinum, vermiculite, and zirconium. However, these minerals are buried deep down the Earth and, hence, surrounded by a large massive waste rock formation. To have access to these minerals and make it available to mankind, the process of mining is usually employed.

Mining is conventionally done through drill and blast operation through which inclined or vertical holes are drilled into the rock formation. Explosives are then used to fragment the rock mass into smaller pieces, thereby creating shock waves in the drilled holes. The blasting event leads to a high chemical reaction which evolves a huge quantity of energy which starts propagating away in a radial direction. Initially, the intensity of the energy is so high that matter near the walls of the blast holes are crushed and displaced radially. However, as the energy intensity decreases, due to geometric spreading, the energy continues to travel through the in situ rock mass as an elastic ground vibration [1, 2]. The unused energy in fragmenting the in situ rock mass also generates other undesirable effects such as flyrock, noise, air overpressure, and backbreak [3,4,5,6]. Moreover, blast-induced ground vibration which is the focus of this study, could cause structural responses and nuisance to humans [7, 8]. In the light of that, it has become a mandatory responsibility of every mining company to monitor the levels of ground vibration during each blast event. This monitoring will provide management of the mine to have the first-hand information on whether the ground vibration levels are below or within the safe levels (standards) set by the Environmental Protection Agency (EPA) of the country.

In general, the vibration monitoring and measurement are done by the use of blasting seismographs. This instrument has a triaxial geophone connected to a processor to collect and analyse the signals. This triaxial geophone contains three mutually perpendicular transducers, each consisting of a spring-loaded moving mass system located within a moving coil to record the three mutually perpendicular components of the motion of the ground particles due to the passage of blasting vibration. These components are: longitudinal (radial) (x), transverse (y), and vertical (z). The particle velocity at a point is the vector sum of the three components at the same instant of time, as shown in Eq. (1) [9]:

$${\text{Particle velocity = }}\sqrt {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}} ,$$
(1)

where \({v_x}\), \({v_y}\), and \({v_z}\) are the particle velocity in the longitudinal, transverse, and vertical directions, respectively. It is worth mentioning that peak particle velocity (PPV) is the most preferable and frequently used indicator for the evaluation of ground vibration. PPV is the velocity of motion of a particle on or in the ground induced by the passing of the blast vibration waves [9].

Over the years, attempt has been made to relate the contributing factors of ground vibration particle motion to the measured PPV. This led to the development of many empirical models presented in Table 1. As these methods have been applied throughout the scientific literature, Table 1 provides only a summary of the modelling method and a description of terms in the related equations.

Table 1 Summary of various PPV empirical predictors
Full size table

The aforementioned empirical models (Table 1) are based solely on two input parameters, namely, distance and maximum charge per delay. Moreover, the empirical predictors are site-specific and, hence, site constants included in the equations (Table 1) must be computed to suit the specific civil or mining industry. Researchers who came up with these models are of the view that the mentioned parameters are the main factors contributing to ground vibration induced by blasting. However, other scholars [2, 19,20,21,22,23 and references therein] argue that the intensity of ground vibration depends on two main groups of parameters, namely controllable and uncontrollable parameters (Table 2) that could be considered and quantified in the development of ground vibration prediction models. The controllable parameters are those that can be changed by the blast engineers, whereas the uncontrollable parameters cannot be changed by the blast engineers [21]. Besides, blast-induced ground vibration is a complex phenomenon with highly nonlinear variable interactions which cannot be adequately modelled using closed-form mathematical equations (Table 1) [24, 25]. In addition, Dindarloo [25] is of the view that lack of generalisability is a common phenomenon inherent in the empirical prediction models (Table 1) because of the number of input parameters considered.

Table 2 Factors that affect ground vibration
Full size table

To address these issues, several research works have attempted to explore soft-computing technology as alternative technique that can accommodate more contributing factors, adequately model nonlinear systems, and produce accurate results. Until now, the notable methods found in the literature for predicting blast-induced ground vibration include artificial neural network (ANN), support vector machines (SVM), fuzzy logic, neuro-fuzzy inference systems (ANFIS), classification and regression tree (CART), hybrid intelligent methods, group method of data handling (GMDH), and genetic expression [25,26,27,28,29,30,31,32,33,34,35,36]. These techniques have been found to outperform the empirical predictors due to their ability to learn, adapt, and generalise well to the data set introduced to them without a priori knowledge of the mathematical association between input and output parameters. Thus, they have the ability to model correctly the complex nonlinear and dynamic system interactions of the blast design, explosive, and geotechnical and geomechanical parameters. The soft-computing techniques also have the capability to tolerate imprecision, uncertainty, approximate reasoning, and partial truth to achieve tractability and robustness on simulating human thinking to match up with reality [37].

It is noteworthy that the literature is replete with the application of ANN and is regarded as the most standard and prominently used soft-computing technique for the prediction of ground vibration. Some important studies are in Refs. [19, 24, 38,39,40,41,42,43,44,45,46,47,48,49,50]. In spite of its wide applicability, ANN suffers from many practical limitations such as slow convergence speed, poor generalisation performance, overfitting problems, and no reasoning capability. It also requires a lot of manual tuning of the model parameters to achieve global optimum. There is also local minima sticking with suboptimal solutions [51,52,53]. Furthermore, there is no proper method to determine the number of hidden neurons other than the sequential trial and error process [54, 55]. It is agreeable that a model should be simple and easily applicable to solving real-world problem. Therefore, to address and overcome some of the weaknesses of the ANN as aforementioned to give better blast-induced ground vibration prediction results and improve convergence, there is the need to explore the other alternative soft-computing methods.

In that respect, this study adopts Gaussian process regression (GPR) for modelling and prediction of blast-induced ground vibration. GPR is a type of Bayesian non-parametric method which can handle the uncertainties in data in a principled manner. It has the possibility to include various kinds of prior knowledge into the model. The number of model parameters, which need to be optimised in the GPR, is less than that of the ANN. Moreover, the tuning of these parameters is carried out using optimiser unlike the ANN which requires human interferences. In addition, the existence of noise in the measured data can be modelled quantitatively in the GPR, thereby having a little impact on the prediction accuracy of the developed GPR model [56]. GPR has been successfully and widely adopted for solving diverse engineering and science related problems [54, 57,58,59,60,61 and references therein]. Although the GPR technique is already in use, quality evaluation of the method in the area of blast-induced ground vibration prediction studies has not been exploited. Furthermore, the GPR technique has not been compared with the benchmark ANN methods to ascertain its reliability in ground vibration prediction. Therefore, taking cognisant of the earlier enumerated mathematical conveniences and benefits provided by the GPR, the paper aims to:

  1. (i)

    investigate the viability of GPR as a novel approach for predicting blast-induced ground vibration using blast event data from an open pit mine in Ghana;

  2. (ii)

    make a comparative study between the developed GPR model and benchmark methods of back-propagation neural network (BPNN), radial basis function neural network (RBFNN), generalised regression neural network (GRNN), and four empirical ground vibration predictors, namely, United State Bureau of Mine (USBM) model, Indian Standard model, Ambraseys–Hendron model, and Langefors and Kihlstrom model.

The rest of the paper is organised as follows. Section 2 presents a brief description of the study area. Section 3 provides a concise overview on the theoretical concept of the proposed GPR approach. Section 4 elaborates on the data set used, post processing, and how the GPR model was developed. In Sect. 5, a description of the statistical evaluators used to ascertain the predictive capabilities of the various models utilised is presented. In Sect. 6, the best GPR model is selected. Prediction results from the selected GPR model are compared with the results produced by BPNN, RBFNN, GRNN, and conventional empirical techniques (USBM model, Langefors and Kihlstrom model, Ambraseys–Hendron model, and Indian Standard model). Section 7 ends the study with conclusions.

2 Brief description of study area

This study was carried out in an open pit mine in Tarkwa, Ghana. The Mine is bounded between latitude 5°16′ North and longitude 1°59′ West in the south-western corner of Ghana. With respect to the map of Ghana, the Mine is situated 89 km northwest from the port of Takoradi and 6.5 km south of Tarkwa [62]. Figure 1 shows the study area.

Fig. 1
figure 1

Study area

Full size image

The open pit mine (study area) has three pits, namely: pit M, pit N, and pit O. Active mining is currently ongoing in pit O. This pit has been subdivided into O South West (OSW), O South East (OSE), O Central West (OCW), O Central East (OCE), and O North (ON). The Mine employs drill and blast techniques to fragment in situ rock formation into suitable rock sizes which are hauled using Volvo AD35, Komatsu HD 465, and CAT 777F rear dump trucks to either the run-of-mine (ROM) or waste dump. Blasting is done with Emulsion EX 7000, 250 g Pentolite booster, a 500 ms down the hole, and either a 25 ms or 42 ms DDX NONEL detonator supplied by Maxam International. The bulk explosives used for the column charge are the Emulsion EX 7000. A 3 m stemming height is applied at the study area with stemming material (gravels) of 15 mm blended with 20 mm to ensure effective confinement. The surface connectors used for the trunk line connections include 17 ms, 42 ms, and 67 ms with an NONEL MS firing method. Three main types of rock formation characterise the study area. These rock types are greenstones, turbidites (metatuffs), and manganiferous horizon. The manganiferous horizon makes up the ore deposit, while the greenstones and metatuffs form the host rocks. The ore mined are the detrital ore, oxide ore, carbonate-oxide ore (carbox), and the manganese carbonate ore. The existence of these different types of ore is due to the different rate of alteration and weathering at the time of formation [62]. The manganese carbonate ore is processed by crushing to aggregates of 100 mm, 40 mm, and 20 mm, which are designated as lumps, logs, and fines, respectively. The crushing plant has a capacity of 300 metric tons per hour. The finished product from the plant is stockpiled and further conveyed to the port in Takoradi, Ghana for export [63]. Monitoring of blast-induced ground vibration is carried out by the environmental department of the Mine. Their monitoring station is located on flat ground near the first house to the mine pit of the nearest community. The monitoring is done using a 3000 EZ Plus Portable seismograph.

3 Methodology

In this section, a concise theoretical concept on the proposed Gaussian process regression will solely be presented. Theories on the benchmark artificial neural network techniques (BPNN, RBFNN, and GRNN), and empirical techniques (USBM model, Langefors and Kihlstrom model, Ambraseys–Hendron model, and Indian Standard model) which have been widely and successfully applied in blast-induced ground vibration prediction will not be treated here. A detailed information on them can be found in [10, 11, 13, 14, 64,65,66,67].

3.1 Gaussian process

A Gaussian process is a stochastic process (a collection of random variables), such that every finite collection of the random variables has a joint Gaussian distribution [68]. A Gaussian process t(x) is parameterised by a mean function m(x) and a covariance function (or kernel) \(k(x,{x^\prime })\) evaluated at points x and \({x^\prime }\). These functions are defined in Eqs. (2) and (3) as follows:

$$m(x)=E(t(x))$$
(2)
$${\text{Cov(}}t(x),t({x^\prime }){\text{)}}=k(x,{x^\prime };\theta )=E((t(x) - m(x))(t({x^\prime }) - m({x^\prime }))),$$
(3)

where \(\theta\) denotes the set of hyperparameters. A Gaussian process t(x) is, hence, expressed in Eq. (4) as follows:

$$t(x)\sim {\text{GP}}(m(x),k(x,{x^\prime })),$$
(4)

where GP stands for gaussian process. This means that the function t(x) is distributed as a Gaussian process with mean m(x) and covariance function \(k(x,{x^\prime })\).

3.1.1 Gaussian process for regression

The goal of every regression problem is to model the dependence of a response variable y on some predictor variables, xi. Each response variable y can be related to an underlying arbitrary regression function t(x) with an additive independent identically distributed Gaussian noise (ε) which represents the noise component from the data. This is expressed in the following equation:

$$y=t(x)+\varepsilon .$$
(5)

The noise ε has zero mean and variance \(\sigma _{{\text{n}}}^{2}\) that is \(\varepsilon \sim N{\text{(}}0{\text{,}}\sigma _{{\text{n}}}^{2}{\text{)}}.\) The Gaussian process represented in Eq. (4) becomes Eq. (6) [69]:

$$t(x)\sim {\text{GP}}(m(x),k(x,{x^\prime })+\sigma _{{\text{n}}}^{2}I),$$
(6)

where I is the identity matrix. Based on the additive nature of the noise \(\varepsilon\) and the marginalization property of GPs, the joint distribution of the training output y at locations X and test outputs \({f_ * }\) at test points \({X_ * }\) is given in Eq. (7) [69]:

$$\left[ {\begin{array}{*{20}{c}} y \\ {{f_ * }} \end{array}} \right]\sim N\left( {\left[ {\begin{array}{*{20}{c}} {m(X)} \\ {m({X_ * })} \end{array}} \right],\left[ {\begin{array}{*{20}{c}} {k(X,X+\sigma _{{\text{n}}}^{2}I)}&{k(X,{X_ * })} \\ {k({X_ * },X)}&{k({X_ * },{X_ * })} \end{array}} \right]} \right).$$
(7)

Conditioning the joint Gaussian prior distribution based on X, y, and \({X_ * }\), the predictive distribution is given in the following equation:

$$p\left( {{y_ * }\left| {X,y,{X_ * }} \right.} \right)\sim N\left( {{{\bar {y}}_ * },\operatorname{var} \left( {{y_ * }} \right)} \right),$$
(8)

where \({\bar {y}_ * }\) (Eq. 9) is the predictive mean and \(\operatorname{var} ({y_ * })\) (Eq. 10) is the predictive variance [69]:

$${\bar {y}_ * }=m({X_ * })+k({X_ * },X){[k(X,X)+\sigma _{{\text{n}}}^{2}I]^{ - 1}}(y - m(X))$$
(9)
$$\operatorname{var} ({y_ * })=k({X_ * },{X_ * }){[k(X,X)+\sigma _{{\text{n}}}^{2}I]^{ - 1}}k(X,{X_ * }).$$
(10)

3.1.2 Covariance function

A covariance function is the central component in a Gaussian process regression model [68]. Therefore, selecting the appropriate covariance function is crucial to the determination of the sample function being modelled. Given that the input points which are closely related are likely to have similar target values; likewise, test point near a training point should have a corresponding target value close to the training point. With this analogy, a test point’s target value can be predicted. This measure of similarity is expressed by the covariance function [70]. There are a number of common covariance functions available in the literature. Some of these include: constant covariance function, linear covariance function, Gaussian noise covariance function, Ornstein–Uhlenbeck covariance function, squared exponential covariance function, Gamma exponential covariance function, Matérn Class of covariance function, Periodic covariance function, Rational quadratic covariance function, and others [68]. However, it is notable in the literature that the squared exponential is the commonly used covariance function [71].

3.1.3 Training a Gaussian process regression model

The parameters of the mean function and covariance (kernel) functions are called the hyperparameters of the Gaussian process [70]. These hyperparameters define the behaviour of the GPR model. To train and formulate a GPR model, all the hyperparameters associated with the mean and covariance function must be learned. This can be done through either optimisation or sampling techniques. However, the widely used approach is to maximise the log marginal likelihood (Eq. 11) [55]:

$$\log p\left( {y\left| {X,\theta } \right.} \right)= - \frac{1}{2}{y^{\rm T}}{\left( {K+\sigma _{{\text{n}}}^{2}I} \right)^{ - 1}}y - \frac{1}{2}\log \left| {K+\sigma _{{\text{n}}}^{2}I} \right| - \frac{n}{2}\log 2\pi ,$$
(11)

where \({y^{\rm T}}\) is the transpose of vector y and \(\theta\) is a vector containing all the hyperparameters.

To maximise the log marginal likelihood, the conjugate gradient method is an efficient gradient-based optimisation algorithm that can be used [72].

4 Model development

In this section, information on the data used, model formulation procedure, and the proposed GPR model developed have been presented.

4.1 Data acquisition and processing

A total of 210 historic blast data set acquired from the Mining and Environmental Department of an Open Pit Mine in Tarkwa, Ghana were used in this study. The data set involved the following parameters: number of blast holes; cooperating charge (kg); distance between blasting point and monitoring station (m); hole depth (m); powder factor (kg/m3); burden (m); spacing (m); peak particle velocity (PPV) (mm/s). For the development of the GPR model and the other investigated methods (BPNN, RBFNN, and GRNN), 6 out of the 8 parameters were used. The input parameters used include the number of blast holes, cooperating charge (kg), distance between blasting point and monitoring station (m), hole depth (m), and powder factor (kg/m3). The output parameter is the PPV (mm/s). The burden and spacing parameters were not considered due to their constant values used by the Mine for each blast. It is worth noting that these parameters were carefully measured. Table 3 outlines the statistical range of measured parameters used in this research. The values for the number of blast holes, cooperating charge (kg), hole depth (m), and powder factor, were computed and obtained from the blasting designs. Global positioning system (GPS) measurements were used to determine the distance between blasting point and monitoring station. Here, the GPS-recorded coordinates between the blasting face and that of the monitoring station were used to calculate the distance. The values of PPV were measured by a 3000 EZ Plus Portable seismograph. It should be noted that the monitoring of the ground vibration was done at the nearest community to the mining pit. Correlation coefficient matrix, which outlines the strength of the relationship between the input parameters [number of blast holes, distance between blasting point and monitoring station, cooperating charge (kg), hole depth (m), and powder factor], and measured PPV is presented in Table 4.

Table 3 Statistical description of the data set
Full size table
Table 4 Correlation coefficient matrix between input parameters and measured PPV
Full size table

To generate good predictions, ANN requires enough data for training. However, if the training data are more than enough, it will cause overfitting, whereby the model cannot perform well with unseen data. Despite the fact that there is no universally accepted ratio for splitting the data, the hold-out cross-validation technique which has been widely and successfully used in ANN modelling was adopted in the present for the data partitioning. Therefore, based on the principle of hold-out cross-validation, it is important that the training data set must be more than the testing set. Here, the entire 210 data set was partitioned into two subsets: training and testing sets. In that regard, the first partition which formed the training data set is made up of 130 data points representing 62% of the entire blasting data. The second partition which formed the testing data set involved the remaining 38% of the data representing 80 points. The training data were purposely selected to represent the entire characteristic of the whole data in the study area. Likewise, the testing data chosen are evenly distributed across the area of study. Here, the training data set was used to build and train the GPR model and the other methods investigated in this study, while the testing data set was used to judge how the model will perform with unseen data.

In the model construction, the training and testing data sets were normalised as part of the pre-processing step. The essence is to ensure constant variability and reduce the impact of variables with high variance to have minimal effects in the model prediction outcomes. This is because the data sets have different range of values with different physical units; Eq. (12) [73] was used to normalise the data into the range [− 1, 1]:

$${M_i}={M_{{\min} }}+\frac{{({M_{{\max} }} - {M_{{\min} }}) \times ({N_i} - {N_{{\min} }})}}{{{N_{{\max} }} - {N_{{\min} }}}},$$
(12)

where Mi is the normalised data, Ni represents the measured blast data, and Nmax and Nmin represent the maximum and minimum values of the measured blast data with Mmin and Mmax values set at − 1 and 1, respectively.

4.2 Proposed GPR model

In this study, a simple mean function with constant, c, was used. For selecting the optimum covariance function for the proposed GPR model, the following covariance functions as expressed in Eqs. (13)–(17) [68] were tried and tested.

  1. i.

    Squared exponential covariance function:

$$k({x_i},{x_j})=\sigma _{{\text{f}}}^{2}\exp \left[ {\frac{{ - {d^2}}}{{2{\ell ^2}}}} \right].$$
(13)
  1. ii.

    Exponential covariance function:

$$k({x_i},{x_j})=\sigma _{{\text{f}}}^{2}\exp \left[ { - \frac{d}{\ell }} \right].$$
(14)
  1. iii.

    Rational quadratic covariance function:

$$k({x_i},{x_j})=\sigma _{{\text{f}}}^{2}\exp {\left[ {1+\frac{{{d^2}}}{{2\alpha {\ell ^2}}}} \right]^{ - \alpha }}.$$
(15)
  1. iv.

    Matérn 3/2 covariance function:

$$k({x_i},{x_j})=\sigma _{{\text{f}}}^{2}\left( {1+\frac{{\sqrt 3 d}}{\ell }} \right)\exp \left( { - \frac{{\sqrt 3 d}}{\ell }} \right).$$
(16)
  1. v.

    Matérn 5/2 covariance function:

$$k({x_i},{x_j})=\sigma _{{\text{f}}}^{2}\left( {1+\frac{{\sqrt 5 d}}{\ell }+\frac{{5{d^2}}}{{3{\ell ^2}}}} \right)\exp \left( { - \frac{{\sqrt 5 d}}{\ell }} \right),$$
(17)

where \(d=\left\| {{x_i} - {x_j}} \right\|\) is the Euclidean distance between point xi and xj, \(\sigma _{{\text{f}}}^{2}\) is the signal variance of function, α is the shape parameter for the rational quadratic covariance, and \(\ell\) is the length scale. Due to the mean function, covariance function, and the noisy observations in data, the hyperparameters \(\theta\) that were optimised in this study include: constant (c), \(\sigma _{{\text{f}}}^{2}\), \(\ell\), α, and \(\sigma _{{\text{n}}}^{2}\). \(\sigma _{{\text{n}}}^{2}\) is the noise variance.

5 Evaluation of model performance

To evaluate the prediction performance of the proposed GPR model to the BPNN, RBFNN, GRNN, USBM model, Langefors and Kihlstrom model, Ambraseys–Hendron model, and Indian Standard model, five statistical indicators namely: mean square error (MSE), root-mean-square error (RMSE), mean absolute error (MAE), correlation coefficient (R), and variance accounted for (VAF). They are mathematically expressed in Eqs. (18)–(23) [6, 74,75,76,77,78,79]:

$${\text{MSE}}=\frac{{\sum\nolimits_{{i=1}}^{N} {{{({O_i} - {P_i})}^2}} }}{N}$$
(18)
$${\text{RMSE}}=\sqrt {{\text{MSE}}} =\sqrt {\frac{{\sum\nolimits_{{i=1}}^{N} {{{({O_i} - {P_i})}^2}} }}{N}}$$
(19)
$${\text{MAE}}=\frac{{\sum\nolimits_{{i=1}}^{N} {\left| {{O_i} - {P_i}} \right|} }}{N}$$
(20)
$$R=\frac{{\sum\nolimits_{{i=1}}^{N} {({O_i} - \bar {O})({P_i} - \bar {P})} }}{{\sqrt {\sum\nolimits_{{i=1}}^{N} {{{({O_i} - \bar {O})}^2} \times \sqrt {\sum\nolimits_{{i=1}}^{N} {{{({P_i} - \bar {P})}^2}} } } } }}$$
(21)
$$VAF=\left[ {1 - \frac{{\operatorname{var} ({O_i} - {P_i})}}{{\operatorname{var} ({O_i})}}} \right] \times 100,$$
(22)

where N is the total number of samples, Oi are the observed values, Pi are the predicted values, \(\bar {O}\) is the mean of the observed values, and \(\bar {P}\) is the mean of the predicted values.

Graphical comparison of the efficiency of the various models in predicting ground vibration was carried out. This was done by plotting the observed PPV against predicted PPV with a 1:1 line, a 95% confidence interval (CI) (Eqs. 23), and 95% prediction interval (PI) (Eq. 24):

$${\text{CI}}=\bar {P} \pm {Z_{\alpha /2}}\frac{\sigma }{{\sqrt n }},$$
(23)

where \(\bar {P}\) is the mean of the predicted values, \(\sigma\) is the population standard deviation, \({Z_{{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}}}\) is the Z value for the desired confidence level \(\alpha\), and n is the number of predicted values. At a 95% confidence interval, \({Z_{{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}}}=1.96\):

$${\text{PI}}={P_i} \pm {t_{(\alpha /2,n - 2)}}{\text{SD}}\sqrt {1+\frac{1}{n}+\frac{{{{({O_i} - \bar {O})}^2}}}{{\sum {{{({O_i} - \bar {O})}^2}} }}} ,$$
(24)

where n is the total number of samples, Oi are the observed PPV values, Pi are the predicted PPV values, \(\bar {O}\) is the mean of the observed PPV values, \({t_{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha {2,n - 2}}} \right. \kern-0pt} {2,n - 2}}} \right)}}\) is the α-level quantile of a t-distribution with n − 2 degrees of freedom, and SD is the standard deviation of the residuals (Eq. 25):

$$s=\sqrt {\frac{{\sum {{{({O_i} - {P_i})}^2}} }}{{n - 2}}} .$$
(25)

6 Results and analysis

In this section, the results of the various techniques considered are presented. In Sect. 6.1, discussion on the GPR models formed based on the different covariance functions used are presented. In Sect. 6.2, a detailed comparison of the proposed GPR model with the other benchmark techniques are presented.

6.1 GPR model results

Using the MSE and R criteria, the comparative results obtained by the various GPR models using the testing data sets are presented in Table 5.

Table 5 Test results of the various Gaussian process regression model
Full size table

From Table 5, it can be observed that the various GPR models had very close R values in the range 0.8300–0.8339 and MSE values falling in the range 0.0245–0.0252. These results confirm that the various GPR models can accurately predict blast-induced ground vibration levels. This is because they all gave R values close to 1 and MSE values very close to 0, indicating good predictive capability. The results further indicated that the covariance functions (Table 5) utilised have similar predictive capabilities. However, in comparison, the GPR model with the Matérn 3/2 covariance function had the highest R value and the second least MSE to the exponential covariance function. The Matérn 3/2 was then selected as the optimum because from Table 5, it is noticeable that the MSE difference between the Matérn 3/2 and exponential covariance function is 0.0001, thus, having very negligible influence on the GPR model outcome. By virtue of the results (Table 5), the selected proposed GPR model valid for blast-induced ground vibration prediction has a constant mean function and Matérn 3/2 covariance function.

6.2 Comparison of GPR model and other investigated techniques

The same data sets (training and testing) applied to the GPR technique were further used to develop models based on the BPNN, RBFNN, GRNN, and empirical predictors (USBM model, Langefors and Kihlstrom model, Ambraseys–Hendron model, and Indian Standard model). The BPNN developed consists of three layers: input, hidden, and output layers. One hidden layer was used in this study, because it has been proven in the literature to have the capability to universally approximate any complex problem [80]. Due to the nonlinearity of the input data to the network, the hyperbolic tangent sigmoid and linear transfer functions were utilised in the hidden and output layers. Training of the BPNN was done using the Levenberg–Marquardt algorithm [81]. The optimum BPNN obtained for this study had five inputs, one neuron in the hidden layer and one output, with the structure [5-1-1]. The developed RBFNN also had three layers namely: input, hidden, and output layer. The Gaussian radial basis function was used in the hidden layer to process the input data to the network. Training was done using the gradient descent algorithm [82]. In the RBFNN, the width parameter of the Gaussian radial basis function and the maximum number of neurons in the hidden layer are the only parameters that need to be adjusted during the training phase. After training, width parameter and maximum number of neurons of 1.6 and 13 which had the lowest MSE and highest R were selected. This led to an optimum RBFNN model having 5 inputs, 13 hidden neurons, and 1 output with structure [5-13-1]. The developed GRNN has four layers: input, pattern, summation, and output layers. The training parameter adjusted to develop the GRNN is the width parameter. In this study, the selected width parameter that gave better MSE and R results was 0.53. With regards to the development of the empirical predictors, site constants were determined. These constants were obtained using regression analysis. Table 6 presents the respective determined model equations for the empirical techniques utilised in this study. The empirical equations utilised in this study have served as benchmark methods over the years when comparison are made with soft-computing methods [38, 44, 83,84,85,86 and references there in]. Hence, the present authors saw it as an opportunity to apply and compare them to the proposed GPR technique. In addition, the empirical predictors applied in this study (Table 6) are being used in the Ghanaian mining industry (study area) for blast-induced ground vibration prediction.

Table 6 Formulated models of the empirical techniques
Full size table

A summary of the statistical comparison of the developed GPR and other investigated methods is presented in Table 7.

Table 7 Developed models’ performance for predicting PPV
Full size table

From Table 7 it can be observed that the GPR-Matérn 3/2 model had the lowest RMSE and MAE of 0.1568 mm/s and 0.1302 mm/s, respectively. These statistical results indicate that the developed GPR-Matérn 3/2 model PPV predictions marginally deviated from the observed PPV. The reason being that the closer the statistical error indicator (MAE and RMSE) is to zero, the better the model could approximate closely the observed data. Comparatively, it can be observed that the soft-computing methods (GPR-Matérn 3/2 model, BPNN, GRNN, and RBFNN) showed a greater strength of relationship and linear dependence between their predicted and observed PPVs than the empirical methods. This is evident in Table 7 where the GPR-Matérn 3/2 model, BPNN, GRNN, and RBFNN had R values above 0.8 while the empirical techniques had below 0.8. This indicates that the predicted PPV values from the soft-computing methods correlate well with the observed PPV than the other models presented in this study. In comparison, the GPR-Matérn 3/2 model was superior to all the methods by exhibiting the highest R value of 0.8338. In Table 7, it can also be observed that, the GPR-Matérn 3/2 approach gave the highest VAF value followed by the other ANN techniques and empirical predictors. In this study, the VAF was used to verify the correctness of the models and how well they could approximate the unseen data set [87]. That is, a model with a VAF value closest to 100% is the most accurate among candidate models. It can, thus, be inferred from Table 7 that the GPR-Matérn 3/2 with the highest VAF value of 68.87% could produce more promising and satisfactory results than the other methods tested. The strength of the GPR-Matérn 3/2 could be attributed to its intrinsic ability to add prior knowledge and specifications about the shape of the model by learning the hyperparameters which are relational to the training and testing data. This helps to capture the uncertainties in the data using the noise variance hyperparameter in the model formulation stage.

Scatter plots along with the regression line, 95% PI, and CI for the various models are presented in Figs. 2, 3, 4, 5, 6, 7, 8, and 9. A critical look at Fig. 2, 3, 4, 5, 6, 7, 8, and 9 shows that the 1:1 line for each model fell in between the 95% CI lines. This means that there is a 95% probability that the true best-fit line for the population data lies within the CI. It can also be observed that 96% of the predicted PPV values fell within the 95% PI lines. This indicates that, when these models are used to predict new PPV values, there is a 95% probability that 96% of the predicted PPV values will fall into the prediction interval. Overall, visual inspection of the various scatter plots revealed that the predictions produced by GPR-Matérn 3/2, BPNN, GRNN, and RBFN had a narrower prediction interval bandwidth than USBM, Ambraseys–Hendron, Langefors–Kihlstrom, and Indian standard model. The comparative results showed that the proposed GPR-Matérn 3/2 model is superior to the benchmark techniques and a promising tool for modelling and predicting blast-induced ground vibration. A comparison of the predicted and measured PPV using only the testing data sets is illustrated as an additional information in Fig. 10. The results demonstrate that predictions through GPR model are more acceptable in comparison with BPNN, GRNN, RBFNN, and empirical models. Furthermore, due to the environmental impacts of blasting operations which leads to severe damage to nearby residence and structure, accurate prediction of ground vibration intensity is key in controlling and minimising its occurrence. For this reason, any predictive model that produces more accurate prediction is of utmost importance to the blast engineer. Therefore, looking at the statistical results related to the proposed GPR and the other methods, this study reveals the GPR as being more accurate and as such should be adopted by the blast engineer in predicting blast-induced ground vibration.

Fig. 2
figure 2

Observed PPV versus predicted PPV by GPR-Matérn 3/2

Full size image
Fig. 3
figure 3

Observed PPV versus predicted PPV by BPNN

Full size image
Fig. 4
figure 4

Observed PPV versus predicted PPV by GRNN

Full size image
Fig. 5
figure 5

Observed PPV versus predicted PPV by RBFNN

Full size image
Fig. 6
figure 6

Observed PPV versus predicted PPV by USBM

Full size image
Fig. 7
figure 7

Observed PPV versus predicted PPV by Ambraseys–Hendron

Full size image
Fig. 8
figure 8

Observed PPV versus predicted PPV by Indian Standard

Full size image
Fig. 9
figure 9

Observed PPV versus predicted PPV by Langefors–Kihlstrom

Full size image
Fig. 10
figure 10

A comparison of the predicted PPV values by predictive models and the measured PPV

Full size image

7 Concluding remarks

In this study, a novel approach to predict ground vibration using the Gaussian process regression (GPR) has successfully been applied. Five different GPR models were developed based on different types of covariance functions using blast data from an Open Pit Mine in Ghana. The GPR models developed and analysed were the squared exponential, exponential, Matérn 3/2, Matérn 5/2, and rational quadratic. Based on the highest correlation coefficient and lowest mean square error criteria, the Matérn 3/2 was the selected covariance of the proposed GPR model. The proposed GPR-Matérn 3/2 model was successfully compared with the other predictive benchmark techniques of BPNN, RBFNN, GRNN, and four empirical techniques (USBM model, Langefors and Kihlstrom model, Ambraseys–Hendron model, and Indian Standard model). The results showed that the GPR-Matérn 3/2 model yields predicted PPV that is better and consistent with the observed PPV than the other methods considered. The GPR-Matérn 3/2 gave the lowest RMSE and MAE of 0.1568 mm/s and 0.1302 mm/s respectively. It also had the highest correlation coefficient of 0.8339, and accounted for the highest percentage of the variability between the measured and predicted PPV with a VAR value of 68.87%. On the basis of the results obtained, it was concluded that the proposed GPR-Matérn 3/2 model has shown promising application potential in blast-induced ground vibration prediction and can serve as suitable substitute to the other benchmark techniques presented in this study. The efficiency of the proposed GPR-Matérn 3/2 approach was attributed to its intrinsic ability to add prior knowledge and specifications about the shape of the model by learning the hyperparameters. However, the underlining limitation of this study is the lack of application of a specific covariance function. That is, one has to iteratively determine the best covariance that will best generalise the data set. This is because the performance of GPR models is highly dependent on the covariance function (or kernel) used. This makes it cumbersome and time consuming. For future studies, automating the selection of the covariance function by combining with metaheuristic algorithm to improve the GPR prediction ability could be explored.