Reply to the Discussion by Saffet Yagiz on “The Effect of Density and Porosity on the Correlation Between Uniaxial Compressive Strength and P-Wave Velocity”

It is acknowledged that Table 1 in the technical note contains equations with mistakes. This table, properly checked, shows that all the equations, except Yagiz’s equation, are correct. Yagiz’s equation was written as UCS = 0.26 V_P^3.453. It should have been written as UCS = 0.258 V_P^3.543.

Table 1 Equations correlating UCS and V_P

With reference to the comments about the comparison of the results obtained by Jamshidi et al. (2018) with those available in the literature (Table 2 and Figs. 1–3 in Discussion), attention needs to be paid to the estimated UCS values obtained with the equations proposed in Jamshidi et al. (2018) and those reported by different researchers.

Table 1 shows the correlations between UCS and V_P given by different researchers as reported in the available literature. The equation given in Jamshidi et al. (2018) is compared with the equations proposed by the authors who have correlated UCS and V_P. It is noted that in our study a good relation in logarithmic terms is found between UCS and V_P with a determination coefficient (R²) equal to 0.82 (Eq. 1).

The difference in the (R²) value obtained in our study and given by other researchers in Table 1 is thought to be due to the difference in the range of physical and mechanical properties of the tested rock types.

The results of a simple regression analysis show R² to be 0.82 between UCS and V_P (Eq. 1). On the other hand, R² based on a multivariate regression analysis indicates that ρ and n have important influence on the correlation between UCS and V_P (Eqs. 2 and 3, respectively).

As a result, the values of R² show that the performance of the multivariate regression equations is higher than that derived from the simple regression equation:

$${\text{UCS}}={\text{131}}.{\text{77 ln }}({V_{\text{P}}}) - {\text{1}}0{\text{48 }}\quad {R^{\text{2}}}=0.{\text{82,}}$$

(1)

$${\text{UCS}}= - \;{\text{127}}.{\text{49 }}+{\text{34}}.{\text{57}}\rho +{\text{ }}0.0{\text{22}}{V_{\text{P}}} \quad {R^{\text{2}}}=0.{\text{86,}}$$

(2)

$${\text{UCS}}={\text{2}}0.{\text{54}} - {\text{3}}.{\text{27}}n+0.0{\text{13}}{V_{\text{P}}} \quad {R^{\text{2}}}=0.{\text{92,}}$$

(3)

where UCS is the estimated value of the uniaxial compressive strength, ρ and n are the density and porosity, respectively, and V_P is the P-wave velocity.

It is noted that, when V_P, ρ and n are taken into consideration for estimating UCS (Eq. 4), no change in R² is obtained.

$${\text{UCS}}= - \;{\text{35}}.{\text{73}}0-{\text{5}}.0{\text{48}}\rho - {\text{3}}.{\text{494}}n+0.0{\text{13}}{V_{\text{P}}}{R^{\text{2}}}=0.{\text{92}}$$

(4)