## 1 Introduction
### Empirical and Statistical Modeling Approach  
Empirical and statistical methods have been proposed for developing flyrock prediction models since the late 1970s [17]. [PERSON] [12] published a study on the development of an empirical formula presented in Equation (1) to predict the maximum distance of throw (\(R_{max}\)) that may result from a blast conducted with a given blasthole diameter (\(d\)). \[R_{max}=260\sqrt[3]{d^{2}}, \tag{1}\]  
The model has been commonly used in the mining and civil construction industries to estimate the potential range of flyrock. The study suggested that an increase in \(d\) leads to a proportionally larger range that the rock fragments can travel. [PERSON] and [PERSON] [18] described an inverse relationship between the ratio \(L\) of stemming length (\(l_{s}\)) to burden (\(B\)) and the distance travelled by the flying fragments (\(R_{f}\)) with a power-law equation between two variables, as shown in Equation (2). \[L=155.2R_{f}{}^{-1.37}, \tag{2}\]  
[PERSON] [19] suggested predictive equations for forecasting the maximum range that flyrock can achieve and the fragmentation size, and this represented a significant effort towards defining the danger zone of blasting. The study included blasts with varying rock explosive properties at different confinement states. It was established that the travel range of a flying rock fragment depends largely on \(d\), the shape factor, and the size of the rock fragment. A probabilistic empirical model was created by [PERSON] et al. [3] to predict flyrock distance using data gathered from 150 blasting operations from the Sungun Copper Mine, Iran. The model facilitated stochastic analysis by considering the variability in blasting parameters. The study identified that higher values of \(l_{s}\), spacing (S), blasthole length (\(l_{bh}\)), \(d\), and powder factor (\(q\)) were correlated with an increased likelihood of flyrock, while an increment in mean charge per blasthole and \(B\) were related to reduced flyrock distance. A comparison of the simulated results with field measurements revealed that the Monte Carlo simulation (MCS) may reasonably estimate the flyrock distance. Further, [PERSON] and [PERSON] [20] introduced a novel approach for enhanced accuracy in flyrock distance estimation. This approach involved correlating the flyrock distance observed using high-speed video cameras in test blasts with the pressure-time history determined by lowering piezoelectric pressure probes into dummy boreholes, behind the blastholes, under actual field conditions. The study examined diverse rock conditions with varied lithologies, rock densities (\(\rho_{r}\)), and jointed beds with adverse orientations. The validated pressure and time combinations were then entered in a projectile motion equation (Equation (3)) and customized to yield the best-fitting flyrock distance (\(R_{fpt}\)) depending on the launch angle (\(\theta\)), mass (\(m\)), and area (\(A\)) of flyrock fragments and \(p_{bc}\) acting upon it over a period (\(\Delta t\)). Here, (\(\Delta p_{t}\)) is the pressure extrapolation for the probe distance in relation to \(B\), and \(g\) is the gravitational acceleration. \[R_{fpt}=\frac{\left[\left(p_{bc}\pm\Delta p_{t}\right)^{2}A^{2}\Delta t^{2} \right]\times sin2\theta}{g\times m^{2}}, \tag{3}\][PERSON] et al. [21] utilized a dataset of 62 blasts to simulate flyrock by developing a predictive model combining Multiple Regression Analysis (MRA) and an MCS for quarry blasts. The MRA preceded the MCS to generate an empirical equation for describing the output-input parameter relationship. Flyrock ejected from blasts was visually observed, and landing sites were recorded using a Global Positioning System (GPS) device. The input parameters included \(B\), \(S\), \(l_{s}\), \(l_{lh}\), \(q\), maximum charge mass per delay (_MCPD_), and rock mass rating (_RMR_). Later, [PERSON] et al.