## 3 Dynamic-Static Conversion Relationship
### Relation Establishment
He found that these two parameters exist regularly and were related to stress. According to the conclusions of the two researchers, there was a specific correlation between the parameters in the empirical formula, which are related to the stress conditions tested. In their published article, [PERSON] et al. [39] studied parameters a and b in type I. They found a high correlation between a and b, with R\({}^{2}\) = 0.91, as shown in \[E_{\rm s}=0.69E_{\rm d}+6.40\quad\quad\text{with R}^{2}=0.75\]3 They proposed a relationship Equation (35) [39] for predicting the static elastic parameters by the parameter b, where b is a parameter associated with rock type. \[E_{\rm s}=0.69E_{\rm d}+6.40\quad\quad\text{with R}^{2}=0.75\]4  
Based on the analysis of the previously published linear correlation between \(E_{\text{s}}\) and \(E_{\text{d}}\) in Table 3, we find that some predictions differ significantly from reality. Some equations may calculate negative values of elastic modulus, so the data with negative constants b in Table 3 are not included in Figure 6, we can observe a strong correlation between the constant parameters a and b. As shown in Figure 6, the correlation is high with R\({}^{2}\) = 0.93. This means that the \(E_{\text{s}}\) depend only on the \(E_{\text{d}}\). The correlation coefficient obtained by this  
\[E_{\rm s}=0.69E_{\rm d}+6.40\quad\quad\text{with R}^{2}=0.75\]5 Correlation between a and b constants [39]. study is higher than that obtained by [PERSON] et al. [39]. The following is the fitting equation between a and b:  
\[E_{\rm s}=0.69E_{\rm d}+6.40\quad\quad\text{with R}^{2}=0.75\]6  
where b is a rock type-related parameter. Further conversion yields the following equation:  
\[E_{\rm s}=0.69E_{\rm d}+6.40\quad\quad\text{with R}^{2}=0.75\]7  
Figure 7 summarizes the curves of the empirical formulae obtained by the previous researchers in Table 1, using the dynamic modulus as the sole independent variable. The relational equation proposed by [PERSON] et al. [32] in Table 1 is not included in the figure because it is necessary to assume a function related to spatial attenuation and a dynamic modulus to plot it. There are also logarithmic correlation equations proposed by [PERSON] and [PERSON] [10], [PERSON] et al. [55], and [PERSON] et al. [33] that are not included in the figure because the bulk density of the specific rock samples must be assumed to plot them. The relational equation proposed by [PERSON] [9] is not included in the figure because the constant (\(\alpha\) and \(\beta\)) does not have a specific value. The figure also does not include the relational equations proposed by [PERSON] and [PERSON] [50] and [PERSON] [52]. The calculated value and the curve deviation are too large. They may be suitable for specific rocks. Therefore, it is not comparable with other curves. \[E_{\rm s}=0.69E_{\rm d}+6.40\quad\quad\text{with R}^{2}=0.75\]8 Correlation between a and b constants. The correlation between static and dynamic moduli of elasticity in the literature is shown in Figure 7, showing significant differences. This analysis also indicates that a larger error implies a significant discrepancy between the prediction and reality. Some equations yield negative values for the elastic modulus, which need to be corrected. Certain empirical equations may be suitable for particular rock types, while others yield significantly inaccurate results. In summary, the predictive empirical formulas listed in Table 1 need to be calibrated for constant values in practical applications, and any calibration may imply new correlations. The correlations in the literature mainly come from uniaxial or triaxial tests on a series of rock types with different elastic moduli. The significant difference in the relationship may be due to certain scholars computing either the secant static elastic modulus or the tangent static elastic modulus. In contrast, others calculate the average static elastic modulus.