## 3 Analysis and Discussion of Results
### UCS Prediction from Selected Bibliography  
As indicated by [PERSON] et al. (2018), the existing simple correlations for limestone from the literature were not suitable for the 'Antofagasta' limestone, which implied the development of a multiple correlation in which more variables (\(v_{p}\), \(\rho_{sat}\) and \(n\)) were introduced to increase the fit of the experimental equation, and to be able to characterize the rock with easily measurable properties. This is due to the fact that the UCS depends both on the index properties (\(v_{p}\), \(\rho_{dry}\), \(n\), Is\({}_{(50)}\), R\({}_{\text{N}}\), etc.), and on the petrofrabrics and internal pore distribution, as can be seen in the values of Table 2. Likewise, the UCS depends on the size of the specimen ([PERSON] et al., 2017; [PERSON] et al., 2018), as well as the \(v_{p}\)([PERSON] et al., 2018; [PERSON] et al., 2018), parameters that must be considered for the scale effect, when applying it to an engineering project.  
In the case of traverting, the same occurs. Table 3 shows the different linear correlations that relate UCS with \(v_{p}\), (Equations (5)-(9)), with \(n\) (Equations (10) and (11)) and \(\rho_{dry}\) (Equation (12)). Equation (13) corresponds to the multiple correlation that includes the three properties. Figure 2 shows how no correlation fits the experimental results (line 1:1), and Table 3 shows the mean values and relative errors. From these average values, it can be concluded that Equations (5) and (11) are the closest to 101.09 and 105.27 MPa, respectively (they have the lowest relative errors in absolute value). It should be noted that in the determination of these equations no traverting specimens were considered, only limestone, marble, dolomitic limestone, dolomite, marble and graveleed limestone.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Analytical Model** & \(\mathbf{R^{2}}\) & **Rock Type** & **Phase** & **Reference** & **Equation** \\ \hline \(v_{\text{P}}=0.0317\text{UCS}+2.0195\) & 0.80 & Carbonate rocks & Dry * & [11] & (5) \\ UCS \(=15\ln\left(1000v_{p}\right)-73\) & 0.86 & Travertime & Dry * & [12] & (6) \\ UCS \(=0.26\chi_{\text{P}}^{253}\) & 0.85 & Travertime, limestone and schist & Dry * & [13] & (7) \\ UCS \(=0.026\text{v}_{\text{P}}-20.207\) & 0.90 & Mary rocks & Dry * & [15] & (8) \\ UCS \(=101.1\ln\left(1000v_{p}\right)-802.81\) & 0.945 & Travertime & Dry * & [16] & (9) \\ UCS \(=143.8\text{e}^{-0.0695n}\) & - & Carbonates and limestones (5 \(<\) n & Dry * & [26] & (10) \\  & & \(<20\%\)) (30 \(<\) UCS \(<\) 150 MPa) & & & \\ UCS \(=135.9\text{e}^{-0.048n}\) & - & Carbonates and limestones (0 \(<\) n & Dry * & [26] & (11) \\  & & \(<20\%\)) (10 \(<\) UCS \(<\) 300 MPa) & & & \\ UCS \(=0.1984\rho_{dry}-362.7\) & - & Basalt, diabase, dolomite, gneiss, granite, limestone, marble, quartztite, rock salt, sandstone, & Dry & [45] & (12) \\  & & & schist, siltstone and tuff & & \\ UCS \(=\frac{1.83}{n}+\frac{0.07}{\rho}+\frac{307}{\
u_{p}}+130\) & 0.54 & Carbonate rocks & Dry * & [14] & (13) \\ \hline \hline \end{tabular}
* The authors do not explicitly indicate saturated or dry conditions, so it is assumed as dry (S\({}_{\text{r}}=0\)).  
\end{table}
Table 2: Existing simple and multiple analytical models between UCS (MPa) and index properties like \(v_{p}\) (km\(\cdot\)s\({}^{-1}\)), \(\rho_{dry}\) (g\(\cdot\)cm\({}^{-3}\)) and \(n\) (%) for traverting and carbonate rocks.