## 3 Dynamic-Static Conversion Relationship
### Relation Establishment
\begin{table}
\begin{tabular}{c c c c c c} \hline
**NO** & **Equation** & **Unit** & **R2** & **Rock Type** & **Reference** \\ \hline
5 & \(E_{\rm s}=1.137E_{\rm d}-9.68\) & GPa & 0.92 & [PERSON] & Belikov et al. (1970) [29] \\
6 & \(E_{\rm s}=1.263E_{\rm d}-29.5\) & GPa & 0.82 &
\begin{tabular}{c} Igeneus– \\ metamorphic \\ \end{tabular} & [PERSON] (1983) [7] \\
7 & \(E_{\rm s}=\alpha E_{\rm d}\,^{\beta}\), \(\alpha\in 0.097\)\(\sim\)0.152, & GPa & 0.96–0.99 & Sandstone–granite & [PERSON] (1987) [9] \\  & \(\beta\in 1.38\sim 1.48\) & & & & \\
8 & \(E_{\rm s}=0.64E_{\rm d}-0.32\) & GPa & 0.84 & All types & Eissa and Kazi (1988) [10] \\
9 & \(E_{\rm s}=0.74E_{\rm d}-0.82\) & GPa & 0.84 & All types & Eissa and Kazi (1988) [10] \\
10 & \(\log_{10}E_{\rm s}=0.7710_{\rm bol}(P_{bulk}E_{\rm d})+0.02\) & GPa & 0.96 & Sedimentary & Eissa and Kazi (1988) [10] \\
11 & \(E_{\rm s}=0.69E_{\rm d}+6.40\) & GPa & 0.75 & All types & McCann and Entwistle (1992) [30] \\
12 & \(E_{\rm s}=1.05E_{\rm d}-3.16\) & GPa & 0.99 & Sedimentary & Christaras et al. (1994) [43] \\
13 & \(E_{\rm s}=0.83E_{\rm d}\) & GPa & - & Goncrele & Neville (1995) [44] \\
14 & \(E_{\rm s}=1.25E_{\rm d}-19\) & GPa & - & Structural design & CP110 (1972) [45] \\
15 & \(E_{\rm s}=0.018E_{\rm d}\,^{2}+0.422E_{\rm d}\) & MPSi & 0.75 & Sedimentary & Lacy (1997) [46] \\
16 & \(E_{\rm s}=0.0293E_{\rm d}\,^{2}+0.4533E_{\rm d}\) & MPSi & 0.74 & Sandstones & Lacy (1997) [46] \\
17 & \(E_{\rm s}=0.0428E_{\rm d}\,^{2}+0.2334E_{\rm d}\) & MPSi & 0.93 & Shales & Lacy (1997) [46] \\
18 & \(E_{\rm s}=0.2807E_{\rm d}\) & GPa & 0.60 & Sedimentary & Brautigam et al. (1998) [47] \\
19 & \(E_{\rm s}=1.153E_{\rm d}-15.2\) & GPa & - &
\begin{tabular}{c} Hard rocks \\ (Es \(>\) 15 GPa) \\ \end{tabular} & Nur and Wang (1999) [48] \\
20 & \(E_{\rm s}=0.8069E_{\rm d}\) & GPa & 0.92 & Composite resin & Helvatgioglu et al. (2006) [49] \\
21 & \(E_{\rm s}=0.7707E_{\rm d}-5854\) & GPa & 0.96 & All types & Mokovciakova and Pandula (2003) [50] \\
22 & \(E_{\rm s}=e^{(0.047E_{\rm d})}\) & GPa & 0.72 & All types & [PERSON] and Soroush (2003) [51] \\
23 & \(E_{\rm s}=0.5087E_{\rm d}-0.6\times 10^{6}\) & Psi & 0.60 & Sandstone & Al-Tahini (2003) [52] \\
24 & \(E_{\rm s}=0.0158(E_{\rm d})\,^{2.74}\) & GPa & - & Shale & Ohen (2003) [53] \\
25 & \(E_{\rm s}=0.541E_{\rm d}+12.852\) & GPa & 0.60 & Limestone & Ameen et al. (2009) [54] \\
26 & \(E_{\rm s}=0.867E_{\rm d}-2.085\) & GPa & 0.96 & Sedimentary & Brotons et al. (2014) [55] \\
27 & \(1.275\log_{10}(P_{bulk}E_{\rm d})-4.714\) & GPa & 0.97 & Sedimentary & Brotons et al. (2014) [55] \\
28 & \(E_{\rm s}=0.932E_{\rm d}-3.421\) & GPa & 0.97 & All types & Brotons et al. (2016) [33] \\
29 & \(\log_{10}E_{\rm s}=\) & & & & \\
30 & \(E_{\rm s}=0.014E_{\rm d}\,^{1.96}\) & GPa & 0.87 & Limestone & Najibi et al. (2015) [56] \\
31 & \(E_{\rm s}=E_{\rm d}\) & GPa & - & Carbonate rocks & [PERSON] et al. (2012) [32] \\
32 & \(E_{\rm s}=0.076(V_{\rm P})\,^{3.23}\) & GPa & - & Shale & Horsrud (2001) [57] \\
33 & \(E_{\rm s}=0.169(V_{\rm P})\,^{3.324}\) & GPa & 0.90 & Limestone & [PERSON] et al. (2015) [56] \\
34 & \(E_{\rm s}=\frac{E_{\rm d}}{3.8\lambda^{-1.08}}\) & - & - & Limestone–marble & [PERSON] et al. (2012) [32] \\ \hline \end{tabular}
\end{table} 
Table 1: Relationship proposed by various authors between static (\(E_{\rm s}\)) and dynamic (\(E_{\rm d}\)) elastic moduli. Where \(\rho_{bulk}\) is the bulk density of the specimen (g/cm\({}^{3}\)); \(V_{\rm p}\) and \(V_{\rm s}\) are expressed in km/s; \(E_{\rm s}\) and \(E_{\rm d}\) are measured in GPa. Here, a, b, c, d, k, f, g, h, \(\alpha\), \(\beta\) are material constants. Since the early 1930s, researchers have been exploring the correlation between static and dynamic elastic properties. Techniques utilizing sound wave propagation have been employed to characterize rocks in mining, petroleum, and geotechnical engineering. Dynamic measurements are frequently employed due to their ease of acquisition and non-destructive. Moreover, few enough rock cores are available for the static method. [PERSON] et al.