## 3 Results and Discussion
### Results of Monte Carlo Simulation  
As mentioned in the previous sections, an RF model and a HHO-RF model were developed for peak particle velocity prediction, and the results obtained show that the HHO-RF model can provide more accurate prediction performance. Hence, the HHO-RF model was used here for Monte Carlo simulation, and four input variables including \(D_{V}\), \(D_{H}\), \(Q_{total}\), and \(Q_{max}\) were considered as the continuous probability distributions (CPD) and two input variables including \(T\) and \(B\) were considered as the discrete probability distribution (DPD) (see Table 2).  
After inputting the probability distribution functions into the MC simulation, a large number of combinations should be randomly selected by using the sampling method. In this regard, two sampling methods including Latin hypercube sampling and simple random sampling are recommended. According to the previous literature, the Latin hypercube sampling method is much better than the simple random sampling method because it can achieve the same calculation performance with fewer simulations. It should be noted that the relationship between input variables should be considered in the MC simulation to guarantee the consistency of input variable combinations in MC simulation and monitoring databases.  
In order to achieve a high level of simulation and probability distribution, 20,000 simulation runs were performed by using the Latin hypercube sampling method, and the relationship of 20,000 randomly selected combinations can be found in Figure 9. It is found that the correlations between input values in MC simulation and blast-induced ground vibration database agree well, so the 20,000 simulation runs developed can be used to analyze the probability distribution of blast-induced ground vibration.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Input Variables** & **Function** \\ \hline \(T\) & Discrete ([0, 25, 50, 75, 100], [0.06, 0.17, 0.34, 0.24, 0.19]) \\ \(B\) & Discrete ([4, 5, 6, 7], [0.17, 0.41, 0.36, 0.06]) \\ \(f\) & Discrete ([5, 6, 7, 8], [0.15, 0.42, 0.18, 0.24]) \\ \(D_{V}\) & Weibull (A, B): A = 65.02, B = 2.62 \\ \(D_{H}\) & Logarithmic (mean, std, dev): mean = 5.06 std. dev = 0.54 \\ \(Q_{total}\) & Weibull (A, B): A = 4613.05 B = 2.15 \\ \(Q_{max}\) & Logarithmic (mean, std. dev): mean = 6.64 std. dev = 0.58 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Probability distribution functions of seven input variables.  
Figure 9: Correlation coefficient of the seven input variables: (a) blast-induced ground vibration database; (b) Monte Carlo simulation.  
After analyzing, the probability distribution of blast-induced ground vibration is shown in Figure 10. As can be seen, the mean value of \(PPV\) in MC simulation results is determined as 0.98 cm/s while the mean value of \(PPV\) is 1.0 cm/s in the database, and the PPV value does not exceed 1.95 cm/s with a probability of 90%. Moreover, the measured \(PPV\) values, the predicted \(PPV\) values by the HHO-RF model, and the Monte Carlo simulation results are also shown in Figure 10. It can be observed that these three cumulative frequency curves are closely related and the curve performed by the MC simulation is smoother. It is also found that these cumulative frequency curves still have some differences, and the differences may be due to the small amount of the monitoring database being too small and the error of the prediction model.  
After obtaining the probability distribution of \(PPV\) values, the engineer can adjust the blasting parameters such as the max charge per delay and burden to control the peak particle velocity and minimize the damage upon the building structure. Moreover, for a case that needs special protection, the engineer is advised to assume the most dangerous situation with the probability of 0.001% to arrange safeguards.