## 2 Background Theory
### Sensitivity Analysis of Each Theory  
Equations (1) and (2) may be rearranged in terms of crack density; crack density is calculated by changing the only selected parameter with constant left parameters. The ratio of change may decrease or increase by up to 100% based on the true value, which can be found in the literature (e.g., [PERSON] et al. [4]). Representative values are summarized in Table 1. Finally, sensitivity analysis was performed using Equation (10):  
\[\text{Sensitivity}=\frac{\text{CR}_{\text{true}}-\text{CR}_{\text{ calculated}}}{\text{CR}_{\text{calculated}}} \tag{10}\]  
where CRtrue is the true crack density obtained from true input parameters, and CRcalculated is the calculated crack density obtained by changing these input parameters. Sensitivity analysis was performed on the variables that form terms in both Equations (1) and (2): compressional wave velocity, shear wave velocity, mass density, Poisson's ratio, bulk modulus of a grain, bulk modulus of the skeleton, bulk modulus of a fluid, shear modulus, and crack aspect ratio. The results of sensitivity testing for these nine parameters are shown in the Figure 3. The most influential factors in Equation (1) were the bulk modulus of the fluid and the aspect ratio, whereas the shear modulus was most influential in Equation (2), as shown by a sensitivity value of around 100. The skeleton bulk modulus has the smallest influence in both Equations (1) and (2). Based on these results, parameters may be listed in order of importance for Equations (1) and (2), as follows: \(\rho>\text{K}_{\text{f}}>\text{K}_{\text{g}}>\gamma\approx\text{V}\approx \text{G}_{\text{S}}>\text{V}_{\text{P}}\approx\text{V}_{\text{S}}>\text{K}_{ \text{sk}}\) and \(\text{G}_{\text{S}}>\gamma\approx\text{K}_{\text{g}}>\text{V}_{\text{P}} \approx\text{V}_{\text{S}}>\rho>\text{V}\approx\text{K}_{\text{f}}\approx \text{K}_{\text{sk}}\), respectively. As each theory uses a different set of background assumptions to derive its Equations, the results of sensitivity analysis also show different patterns; thus, it is non-trivial to physically connect both theories. This limitation was solved by establishing a relationship between crack density and aspect ratio based on the measured values in each Equation. Consequently, crack density or aspect ratio can be calculated by solving two simultaneous Equations.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Input Parameter** & **Symbol** & **Value** & **Unit** \\ \hline Compressional wave velocity & \(\text{V}_{\text{P}}\) & 3837 & m/s \\ Shear wave velocity & \(\text{V}_{\text{S}}\) & 2213 & m/s \\ Mass density & \(\gamma\) & 2800 & kg/m\({}^{3}\) \\ Poisson’s ratio & \(\
u\) & 0.1 & - \\ Bulk modulus of grain & \(\text{K}_{\text{g}}\) & 15,000,000,000 & Pa \\ Bulk modulus of skeleton & \(\text{K}_{\text{sk}}\) & 7,780,000 & Pa \\ Bulk modulus of fluid & \(\text{K}_{\text{f}}\) & 2,180,000,000 & Pa \\ Shear modulus of grain & \(\text{G}_{\text{S}}\) & 13,000,000,000 & Pa \\ Aspect ratio & \(\rho\) & 0.19 & - \\ \hline \end{tabular}
\end{table}
Table 1: True value for sensitivity analysis obtained by [PERSON] et al. [4].  
Figure 3: Results of sensitivity testing for (**a**) compressional wave velocity, (**b**) shear wave velocity, (**c**) mass density, (**d**) Poisson’s ratio, (**e**) bulk modulus of a grain, (**f**) bulk modulus of a skeleton, (**g**) bulk modulus of a fluid, (**h**) shear modulus, and (**i**) aspect ratio.