## 2 Materials and Methods
### Gene Expression Programming (GEP)  
To obtain a mathematical relation, the method of GEP has been developed for the prediction of PPV. This method was firstly introduced by [PERSON] [78]. Recently, the application of this method has increased in civil and mining engineering [7; 67; 79; 80; 81; 82; 83; 84]. With the use of different mathematical operators and linear functions (e.g., +, -, sin, tan, cos, sqrt, x\({}^{2}\), and x\({}^{3}\)), this method forms a mathematical relation between the input and output variables. In the following, a summary of the GEP modelling process is presented:  
1. On the basis of population number, a certain number of chromosomes are produced in a random way.
2. The chromosomes of the initial population are denoted as mathematical equations.
3. Each chromosome's fitness is measured based on the fitness function (coefficient of determination, R\({}^{2}\), or root mean square error (RMSE)). If the stopping criteria are not met, the best of the first generation is chosen using the roulette wheel method.
4. The genetic operators, which are known as the core of the GEP algorithm, are applied to the rest of chromosomes for the purpose of creating modified individuals.
5. At this step, the chromosomes will start creating the next generation. This process is iterated for a certain number of generations. More information about this method can be found in the work of other researchers [78; 84; 80; 83].  
Various models of GEP were applied in which non-normal distributions of data were used. A total of 80% of the data was allocated to the training section and 20% to the testing section. Regarding the use of two statistical indicators of R\({}^{2}\) and RMSE [85; 86; 87; 87] for determining the best model, a scoring system was used, as introduced by [PERSON] et al. [88]. The highest and lowest values of R\({}^{2}\) gain the highest and lowest score, respectively, while the situation is reversed for RMSE. For example, in the training section, values of R\({}^{2}\) for models 6 and 2 are 0.7283 and 0.6927, respectively, and the highest (10) and lowest (1) scores are obtained for them. However, for the RMSE of the training, the lowest error is related to model 4, and thus obtains the highest score (10), and model 10 also obtains the lowest score because of its maximum error value. This method of ranking is applied in different sections, and at the end, the total score of each row is placed into the last column. As a result, model number 6 obtains the highest score and is introduced as the best model for predicting PPV (Table 3). As a result, R\({}^{2}\) of 0.7283 and 0.6823 and RMSE of 4.2814 and 4.0344 for training and testing, respectively, show suitable levels of prediction for developing the GEP model.  
The best PPV equation obtained by the GEP model is presented in Equation (6). This equation is a summation of Equations (1) to (5), which are presented as follows:  
\[Gen1=d(1) \tag{1}\]  
\[Gen2=\cos(\sin(\sqrt{d(0)}))\times\frac{(-22.347)\times(\sin(d(2))}{d(0)+d(1)} \tag{2}\]  
\[Gen3=37.028 \tag{3}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Symbol** & **Unit** & **Range** \\ \hline Stemming & ST & m & 1.4–4 \\ Burden to spacing ratio & BS & - & 0.42–0.91 \\ Maximum charge per delay & C & kg & 69.79–309.09 \\ Powder factor & PF & kg/m\({}^{3}\) & 0.24–0.98 \\ Distance & D & m & 65–640 \\ Peak particle velocity & PPV & mm/s & 2.27–37.44 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Range, unit and symbol of the measured parameters in the sites.  
\[Gen4=\frac{d(4)}{-14.041} \tag{4}\] \[Gen5=\tan(\cos(\frac{\cos(d(3))}{\sin(d(0))}-\frac{d(4)}{d(2)})+\sin( d(3)+7.647)))\] (5) \[Y=Gen1+Gen2+Gen3+Gen4+Gen5 \tag{6}\]  
where, the variables of \(d(0)\), \(d(1)\), \(d(2)\), \(d(3)\), \(d(4)\), and \(Y\) are Burden to Spacing, Stemming, Powder Factor, Max charge Per Delay, Distance and PPV, respectively.  
Figures 3 and 4 show the results of PPV prediction using Equation (6), for training and testing, respectively.