## 4 Analysis
### Nonlinear Multiple Regression Analysis
The equation for \(P_{L}\) derived by nonlinear multiple regression analysis is presented in Equation (5) and, also has a higher \(R^{2}\) of 0.46 compared with the corresponding simple regression equation (Figures 8a and 10a):  
\[E_{m}=\ -35.1588+0.11367(N_{60})^{1.2859}+136.3515(\frac{VR}{\sigma_{v}^{r}/P_{a} })^{-1.1625}\ (\text{MPa}),\ R^{2}=\ 0.76 \tag{4}\]  
\[P_{L}=\ -9.0592+0.0001(N_{60})^{1.8969}+18.4856(\frac{VR}{\sigma_{v}^{r}/P_{a} })^{-0.3767}\ (\text{MPa}),\ R^{2}=\ 0.46 \tag{5}\]  
For evaluation, the results obtained by the empirical Equations (4) and (5) derived in this study were compared with the results of equations derived in relevant previous studies presented in Table 2, as shown in Figure 11. The types of soils for the equation presented in Table 2 are different from that used in this study. However, unfortunately, there are very few studies for estimating the relationship between the results of PMT and SPT on weathering granite. Therefore, in this study, the most similar studies performed on sandy material among the existing studies were compared with the Equations (4) and (5). The equations in the previous studies used a single variable, SPT-\(N_{60}\), and can thus be only used to estimate \(E_{m}\) or \(P_{L}\) for a certain SPT-\(N_{60}\). However, the equations derived in this study use an additional variable, \(VR\), which is a measure of the degree of weathering and can thus be used to estimate \(E_{m}\) and \(P_{L}\) from SPT-\(N_{60}\). Furthermore, the present empirical equations derived by nonlinear multiple regression can more accurately reproduce the PMT results for South Korean granite compared with the equations suggested by the other studies that are presented in Table 2.  
Figure 10: Correlations between \(P_{L}\) and _normalized VR_ determined by simple regression analyses using an (**a**) allometric model and a (**b**) linear model. Although the equation proposed by [PERSON] and [PERSON] [6] was based on the analysis of weathered granite in Hong Kong, which is similar to the target stratum of this study, it underestimates \(E_{m}\) and \(P_{L}\) for the weathered granite of the present study. This is because [PERSON] and [PERSON] [6] derived the equations using data that contained SPT-\(N_{60}\) values of \(\leq\)130, as opposed to those with SPT-\(N_{60}\) values of 61-644 that were used in the present study. The equations proposed by [PERSON] and [PERSON] [8] were based on the results of SPTs and PMTs conducted on the medium to very dense sandy soils with SPT-\(N_{60}\leq 100\) and also underestimate the PMT results for the weathered granite examined in the present study. However, [PERSON] and [PERSON] [9] obtained their empirical equations based on data for dense to very dense silty sands (SPT-\(N_{60}<50\)), with the predictions of \(E_{m}\) for weathered granite comparable to those of the present corresponding equation; however, their predicted \(P_{L}\) are higher than those obtained in the presented study. Regarding the similarity for \(E_{m}\), the \(E_{m}\) values for the specific samples used by [PERSON] and [PERSON] [9], where SPT-\(N_{60}\geq 35\), were approximately 30% larger than those for the samples used by [PERSON] and [PERSON] [8]. Therefore, a local correlation based on the local rock characteristics is important for predicting the engineering properties of geo-materials. The proposed equations of \(E_{m}\) and \(P_{L}\) are intended for use in the geotechnical survey of weathered granite in South Korea.