## 5 Prediction of Weathering Based on the Freeze-thaw Process by Rock Type  
High correlation between \(\mathrm{A_{bs}}\) and \(\mathrm{V_{p}}\) in Figure 7 can be implemented in development of an equation predicting the attenuation of weathering due to physical changes in relation to freeze-thaw. The arrows towards to the upper left show freeze-thaw weathering progresses of the rocks in Locations A and B. Clear patterns of changes in \(\mathrm{A_{bs}}\) and \(\mathrm{V_{p}}\) with increasing freeze-thaw cycle enable estimation of rock weathering grade with changes of freeze-thaw cycles (or time if average freeze-thaw cycle number per year is known.) Equations (1) and (2) present changes of \(\mathrm{A_{bs}}\) and \(\mathrm{V_{p}}\) with increasing freeze-thaw cycle, which are the results from the P [27] of two companion papers. These equations can be used to calculate the average amount of weathering attenuation cycles for each location, as different coefficients are given in Table 4 for each location. \[A_{bs}=\alpha N+A_{bs,N0} \tag{1}\]  
where \(\alpha\) is the gradient of the linear relationship between \(\mathrm{A_{bs}}\) and the freeze-thaw cycle number N and \(\mathrm{N_{bs,N0}}\) is the \(\mathrm{A_{bs}}\) values when N = 0. \[V_{p}=\beta N+V_{p,N0} \tag{2}\]  
where \(\beta\) is the gradient of the linear relationship between \(\mathrm{V_{p}}\) and N and \(\mathrm{V_{p,N0}}\) is the \(\mathrm{V_{p}}\) values when N = 0. From the given Equations (1) and (2), the remaining number (\(\Delta N_{remaining}\)) of freeze-thaw cycle from current and target freeze-thaw cycles (\(N_{Current}\) and \(N_{Target}\)) using \(\mathrm{A_{bs}}\) and \(\mathrm{V_{p}}\) can be estimated using the following equations, respectively:  
\[\Delta N_{remaining,A_{bs}}=N_{Target}-N_{Current}=\frac{A_{bs,Target}-A_{bs, Current}}{\alpha} \tag{3}\]  
where \(\Delta N_{remaining,A_{bs}}\) is \(\Delta N_{remaining}\) estimated in using \(\mathrm{A_{bs}}\) and \(\mathrm{A_{bs,Target}}\) and \(\mathrm{A_{bs,Current}}\) are the target and current \(\mathrm{A_{bs}}\) values. \[\Delta N_{remaining,V_{p}}=\frac{V_{p,Target}-V_{p,Current}}{\beta} \tag{4}\]  
where \(\Delta N_{remaining,V_{p}}\) is \(\Delta N_{remaining}\) estimated in using \(\mathrm{V_{p}}\) and \(\mathrm{V_{p,Target}}\) and \(\mathrm{V_{p,Current}}\) are the target and current \(\mathrm{V_{p}}\) values. Equation (5) shows the relationship between \(\mathrm{A_{bs}}\) and \(\mathrm{V_{p}}\) in Figure 7 with different coefficients of \(\gamma\) in Table 4. Equation (5) can be used to convert \(\mathrm{A_{bs}}\) to \(\mathrm{V_{p}}\) or vice versa when one property is missing. However, conversion one property from the other property using Equation (5) may induce higher uncertainty in estimation of \(\Delta N_{remaining}\). \[A_{bs}=\gamma V_{p}+A_{bs,V_{p0}} \tag{5}\]  
where \(\gamma\) is the gradient of the relationship between \(\mathrm{A_{bs}}\) and \(\mathrm{V_{p}}\) and \(\mathrm{A_{bs,V_{p0}}}\) is the y-intercept when \(\mathrm{V_{p}}\) = 0. Using Equations (1) and (2) to quantify physical changes in each rock type due to freeze-thaw weathering, prediction of the duration (remaining time) of freeze-thaw to change weathering grades based on \(\mathrm{V_{p}}\) for rocks from Location A was found to be shorter than that of rocks from Location B. This means that physical properties of rocks, such as \(\mathrm{A_{bs}}\) and \(\mathrm{V_{p}}\), show attenuation effects due \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Location** & \(\alpha\) & \(\beta\) & \(\gamma\) & \(\mathrm{A_{bs,N0}}\) & \(\mathrm{V_{p,N0}}\) & \(\mathrm{A_{bs,V_{p0}}}\) \\ \hline A & 0.000236 & \(-\)1.1223 & \(-\)0.0002 & 0.1444 & 6155.3 & 1.3307 \\ B & 0.000169 & \(-\)0.1398 & \(-\)0.0008 & 1.2815 & 4170.1 & 4.8151 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Coefficients and reference parameters of fitting parameters of the linear regression equations from (1) to (5).