## 3 Results and Discussion  
Linear regression analyses were performed to assess the relationships between the ultrasonic P-wave velocities (V\({}^{\mathrm{n}}\)\({}_{\mathrm{p}}\), V\({}^{\mathrm{d}}\)\({}_{\mathrm{p}}\), and V\({}^{\mathrm{s}}\)\({}_{\mathrm{p}}\)) and mechanical properties (UCS, PLI\({}^{\mathrm{A}}\), PLI\({}^{\mathrm{D}}\), ITS, and SHV) of the limestone samples. The numbers and diameters of the rock samples for each test are given in Table 1. The best-fit line equation and the correlation coefficient, R, were also obtained, as displayed in Table 1.  
In Figures 1-5, the best-fit lines for each data set are shown. In all cases, linear regression was considered as the foremost representative for the best-fit relationships. The P-wave velocity values were highly scattered with respect to all mechanical properties of limestone, as shown in Figures 1-5. This trend could be due to the anisotropic and nonhomogeneous nature of the rock samples and the influence of the sample diameter. The relationships between the UCS and V\({}^{\mathrm{n}}\)\({}_{\mathrm{p}}\), V\({}^{\mathrm{d}}\)\({}_{\mathrm{p}}\), and V\({}^{\mathrm{s}}\)\({}_{\mathrm{p}}\) for various sample diameters are displayed in Figure 1a-c. The correlation coefficient values have a range of 0.04-0.95 (Table 1). The maximum R-value was 0.95, indicating a strong correlation between the UCS and V\({}^{\mathrm{n}}\)\({}_{\mathrm{p}}\) for a rock sample diameter of 63.2 mm. In contrast, no correlation was observed for V\({}^{\mathrm{d}}\)\({}_{\mathrm{p}}\) at the same diameter. This result was unexpected and may be due to either invisible natural features of the tested samples or human error during the experiments. In addition, the International Society for Rock Mechanics (ISRM) [48] standard states that the specimen diameter for the UCS test should not be less than the NX core size (approximately 54 mm). Therefore, the 53.9-mm sample diameter could be considered to provide the best correlation between the P-wave velocities and UCS. Figure 2a-c displays the relationships between PLI\({}^{\mathrm{D}}\) and V\({}^{\mathrm{n}}\)\({}_{\mathrm{p}}\), V\({}^{\mathrm{d}}\)\({}_{\mathrm{p}}\), and V\({}^{\mathrm{s}}\)\({}_{\mathrm{p}}\) for different sample diameters. The R-values ranged from moderate to strong (0.58-0.98) (Table 1). In general, the correlations between PLI\({}^{\mathrm{D}}\) and the P-wave velocities were relatively strong. The maximum R-value was observed for PLI\({}^{\mathrm{D}}\) and V\({}^{\mathrm{s}}\)\({}_{\mathrm{p}}\) at a sample diameter of 38.1 mm. The values of PLI\({}^{\mathrm{A}}\) are plotted as a function of V\({}^{\mathrm{n}}\)\({}_{\mathrm{p}}\), V\({}^{\mathrm{d}}\)\({}_{\mathrm{p}}\), and V\({}^{\mathrm{s}}\)\({}_{\mathrm{p}}\) for various sample diameters in Figure 3a-c. The correlations between PLI\({}^{\mathrm{A}}\) and the P-wave velocities exhibit strong variations, ranging from negative to positive (weak to strong) with R-values of \(-0.29\) to 0.92 (Table 1). The maximum and minimum correlation coefficient (R) values were acquired for V\({}^{\mathrm{n}}\)\({}_{\mathrm{p}}\) and V\({}^{\mathrm{s}}\)\({}_{\mathrm{p}}\) at sample diameters of 47.7 and 63.2 mm, respectively. For different sample diameters, the relationships between the ITS and V\({}^{\mathrm{n}}\)\({}_{\mathrm{p}}\), V\({}^{\mathrm{d}}\)\({}_{\mathrm{p}}\), and V\({}^{\mathrm{s}}\)\({}_{\mathrm{p}}\) are presented in Figure 4a-c. Except for the 24.9-mm diameter samples, the R-values indicated moderate to strong correlations, with a range of 0.62-0.83 (Table 1). The minimum and maximum R-values were observed for V\({}^{\mathrm{s}}\)\({}_{\mathrm{p}}\) and the V\({}^{\mathrm{n}}\)\({}_{\mathrm{p}}\) at sample diameters of 24.9 and 63.2 mm, respectively. Figure 5a-c illustrates the relationships between the SHV and V\({}^{\mathrm{n}}\)\({}_{\mathrm{p}}\), V\({}^{\mathrm{d}}\)\({}_{\mathrm{p}}\), and V\({}^{\mathrm{s}}\)\({}_{\mathrm{p}}\) for various sample diameters.