## 4 Grey Relational Analysis of Disturbance Sources in Zhanjiang Formation Structural Clay
### Grey Relational Analysis  
Assume there is a reference sequence \(X_{0}=\left\{x_{0}(k)\,,\,k=1\,,\,2\,,\,\cdots n\right\}\) and a comparative sequence \(X_{i}=\left\{x_{i}(k)\,,\,k=1\,,\,2\,,\,\cdots n\right\}\), then the relational coefficient at point \(k\) between sequence \(X_{0}\) and \(X_{i}\) is denoted as \(\xi_{k}\).  
\[\xi_{k}=\frac{\min_{i}\min_{k}\left|x_{i}(k)-x_{0}\left(k\right)\right|+\rho \max_{i}\max_{k}\left|x_{i}(k)-x_{0}\left(k\right)\right|}{\left|x_{i}(k)-x_{ 0}\left(k\right)\right|+\rho\max_{i}\max_{k}\left|x_{i}(k)-x_{0}\left(k\right) \right|} \tag{3}\]  
In the formula, \(\rho\) is the resolution coefficient, which lies between 0 and 1, and is typically set at 0.5.  
Equation (3) is simplified to Equation (4).  
\[\xi_{i}(k)=\frac{\Delta(\min)+\rho\Delta(\max)}{\Delta_{i}(k)+\rho\Delta(\max)} \tag{4}\]  
In the equation, \(\Delta(\min)\) denotes the minimum second-order difference, and \(\Delta(\max)\) signifies the maximum second-order difference.  
Since the relational coefficient represents the correlation degree between the comparative sequence and the reference sequence at each moment (i.e., each point on the curve), there is not just a single value, and the information is too dispersed for a holistic comparison. Therefore, we consolidate the relational coefficients at each moment (i.e., each point on the curve) into a single value, namely by calculating their average value. This average value  
\begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{**Experiment No.**} & \multicolumn{2}{c}{**Duration**} & \multicolumn{2}{c}{**Frequency**} & \multicolumn{2}{c}{**Experimental Results**} \\  & **(min)** & **(Hz)** & & **Disturbance Degree** & **Disturbance Degree** \\  & **1 (t)** & **2 (f)** & & \(RD_{q}\) & \(RD_{s}\) **(\%)** \\ \hline \multirow{2}{*}{**Horizontal mean**} & \(k_{1}\) & 0.26 & 0.317 & 31.767 & 31.633 \\ \cline{2-6}  & \(k_{2}\) & 0.377 & 0.397 & 35.767 & 36.767 \\ \cline{2-6}  & \(k_{3}\) & 0.507 & 0.430 & 42.333 & 41.467 \\ \hline \multirow{2}{*}{**Range \(R_{j}\)**} & \multicolumn{2}{c}{\(R_{t}\)} & \multicolumn{2}{c}{0.247} & 10.567 \\ \cline{2-6}  & \multicolumn{2}{c}{\(R_{f}\)} & \multicolumn{2}{c}{0.113} & 9.833 \\ \hline \end{tabular}
\end{table}
Table 7: Cont.  
serves as a quantitative representation of the correlation degree between the comparative sequence and the reference sequence, denoted as the correlation degree \(\gamma_{i}\).  
\[\gamma_{i}=\frac{1}{n}\sum_{i=1}^{n}\xi_{i}(k) \tag{5}\]  
For the results of the disturbance degree, grey relational analysis is conducted, taking the disturbance degree as the reference sequence, and the duration of factor A and the frequency of factor B as the comparative sequences. Since the two factors have different physical meanings, resulting in different units of measurement, dimensionless processing is first carried out for ease of comparison. Then, the relational coefficients are calculated using Equation (4), with \(\Delta(\min)=0\) and \(\Delta(\max)=1.3900\). Finally, the grey relational degrees are calculated using Equation (5) and are denoted as \(\gamma_{1}=0.6769\) and \(\gamma_{2}=0.6072\), respectively.  
Through the calculation of the relational degrees, it is found that both the frequency of factor A and the duration of factor B have a significant impact on the disturbance. According to the principles of grey relational analysis, the larger the relational coefficient, the greater the impact. Therefore, comparing the influencing factors \(\gamma_{1}\) (duration) and \(\gamma_{2}\) (frequency), the disturbances caused by duration are significantly more pronounced than those caused by frequency. This is consistent with the results analyzed from the previous orthogonal experimental analysis.