## 4 Analysis
### Nonlinear Multiple Regression Analysis  
To derive the equations for estimating \(E_{m}\) and \(P_{L}\) based on the chemical weathering index (\(VR\)) determined by the XRF analysis of the collected samples and the results of the PMTs and SPTs conducted at the three considered sites, nonlinear multiple regression analysis was performed using the following three steps:  
1. Deduction of the relationships between \(E_{m}\) and SPT-\(N_{60}\) and between \(P_{L}\) and SPT-\(N_{60}\) by simple regression analyses;
2. Deduction of the relationships between \(E_{m}\) and _normalized_\(VR\) and between \(P_{L}\) and _normalized_\(VR\) by simple regression analyses;
3. Nonlinear multiple regression analysis using the relationships deduced in steps (1) and (2) above. In the implementation of the first step, the relationship between the PMT results (\(E_{m}\) or \(P_{L}\)) and SPT-\(N_{60}\  
In the second step, the procedure of the first step was applied to _normalized VR_. \(E_{m}\) was decreased with increasing _normalized VR_, with a higher correlation coefficient obtained for the allometric model (\(R^{2}\) = 0.67) than for the linear model (\(R^{2}\) = 0.31) (Figure 9). The correlation of \(P_{L}\) with _normalized VR_ was also higher for the allometric model compared with the linear model, although the \(R^{2}\) values (0.33 and 0.18, respectively) were lower than that for the correlation between \(E_{m}\) and _normalized VR_ (Figure 10). Subsequently, the nonlinear multiple regression analysis models of \(E_{m}\) and \(P_{L}\) were established by combining the allometric model, which produced the better correlations between the PMT results (\(E_{m}\) and \(P_{L}\)) and SPT-\(N_{\rm 60}\) and _normalized VR_. The obtained models are:  
\[E_{m}=\ f(N_{\rm 60^{\prime}},\frac{VR}{\sigma_{\rm p}^{\prime}/P_{a}})\ =\ a_{1}+a_{2}(N_{\rm 60})^{a_{3}}+a_{4}\!\left( \frac{VR}{\sigma_{\rm p}^{\prime}/P_{a}}\right)^{a_{5}}(a_{n}=\ \mbox{const.})\ \mbox{and} \tag{2}\]  
\[P_{L}=\ f(N_{\rm 60^{\prime}},\frac{VR}{\sigma_{\rm p}^{\prime}/P_{a}})\ =\ b_{1}+b_{2}(N_{\rm 60})^{b_{3}}+b_{4}\!\left( \frac{VR}{\sigma_{\rm p}^{\prime}/P_{a}}\right)^{b_{5}}(b_{n}\!=\mbox{const.}). \tag{3}\]  
Figure 8: Correlations between \(P_{L}\) and SPT-\(N_{\rm 60}\) determined by simple regression analyses using an (**a**) allometric model and a (**b**) linear model. Figure 9: Correlations between \(E_{m}\) and _normalized VR_ determined by simple regression analyses using an (**a**) allometric model and a (**b**) linear model. The results derived by nonlinear multiple regression analysis revealed improved correlations compared with those from the equations derived by simple regression analysis. Even though the obtained slight increase in the coefficient of determination was lower than expected, this study is meaningful in that it is demonstrates that improved prediction of the \(E_{m}\) and \(P_{L}\) values of weathered granite can be obtained by additionally considering the \(VR\), chemical weathering index that represents the physical properties of weathered granite. The equation for \(E_{m}\) as a function of SPT-\(N_{60}\) and _normalized VR_ is presented in Equation (4), for which \(R^{2}=0.76\), which is larger than that for the simple regression analysis equation (Figures 7a and 9a).