## 5 Methodology
### Analyais for Real-Time Compaction Quality Heterogeneity Using i-GAM  
The current evaluation method for compaction quality of earth-rock dam has ignored the inherent inhomogeneity of the evaluation indices within the working face, which may lead to uneven settlement  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Importance Degree** & **Implication** \\ \hline
1 & The two elements are equally important \\
3 & The former is slightly more important than the later \\
5 & The former is rather more important than the latter \\
7 & The former is strongly more important than the latter \\
9 & The former is absolutely more important than the latter \\
2, 4, 6, 8 & Indicate the intermediate value of the above adjacent judgments \\ \hline \hline \end{tabular}
\end{table}
Table 2: Importance degree and implication for elements in i-AHP system.  
and arch effect of the dam. Based on enough information collected by the RTCM system, \(i\)-GAM can help to quantify the inhomogeneity of real-time compacted quality, which can be expressed as:  
\[\gamma(h)=\frac{1}{2}E[Z(x)-x(x+h)]^{2}, \tag{22}\]  
where \(\gamma(h)\) denotes the semivariance function, \(h\) represents the lag distance (m), \(Z(x)\) and \(Z(x+h)\) denotes the evaluation indices collected at distance \(x\) and \((x+h)\), respectively. However, due to the difficulty of directly calculating the semivariance, Equation (23) is introduced to obtain them by sample points.  
\[\left\{\begin{array}{l}\gamma(h)=\frac{1}{2N(h)}\sum_{i=1}^{N(h)}[Z(x)-Z(x+h )]^{2}\\ h=num\cdot lag\\ num=int\Big{(}\frac{dis}{lag}+0.5\Big{)}+1\end{array}\right., \tag{23}\]  
where \(N(h)\) represents the number of data pairs, \(lag\) denotes the lag class distance interval and doesn't exceed 1/2 of the maximum distance of data pairs (m). \(num\) denotes lag class, \(dis\) is the actual distance between each data pair (m), and int is the integer function.  
The dotted pair data (\(h\), \(\gamma(h)\)) needs to be marked in cartesian coordinates, afterwards the semivariogram can be obtained through curve-fitting as shown in Figure 4. The most commonly used fitting model is the exponential model as Equation (24) [27].  
\[\gamma(h)=C\Big{(}1-e^{-\frac{3h}{\alpha}}\Big{)}, \tag{24}\]  
where \(C=\lim\limits_{h\rightarrow+\infty}\gamma(h)\) represents the sill, which indicates the maximum difference of evaluation indices in space. \(a\) denotes the range, which represents the minimum mean distance to reach the plateau sill. A lower sill and larger range show stronger compaction uniformity and geospatial continuity.  
In Reference [27], the coefficient of the semivariogram (_Cova_) index was proposed to quantify the geospatial heterogeneity of materials. Results show that the _Cova_ index has a close value and trend with the coefficient of variance (_Cov_) index of material evaluation indices. However, we have deduced that the above two indexes theoretically have the same value, which is elaborated in the Appendix A. Therefore, we can infer that the small difference between _Cova_ and _Cov_ in Reference [27] is caused by the error of curve-fitting. In the demonstration, Equation (25) is obtained.  
\[C=std^{2}, \tag{25}\]  
where \(std\) is the standard deviation of evaluation indices. Therefore, in the process of curve-fitting of semivariance function in this paper, the value of sill \(C\) in the exponential model can be predefined  
Figure 4: Relationship between semivariance function and covariance function.  
rather than being fitted by a mathematical method, which will surely improve the accuracy of the fitting model.