## 3 Methodology
### Safety Distance Model  
The establishment of the anchoring radius model was based on the full consideration of the circular motion characteristics of the anchoring ship. However, due to the interaction between anchoring ships, the safety distance between two anchoring ships was taken into consideration so as to avoid the safety interference between multiple ships. Figure 5 simulates the motion trajectories of two anchoring ships. Figure 5 shows the motion dynamics of the two anchoring ships in synchronous motion, and the distance between them satisfies Equation (6) in this state. \[\left\{ \begin{array}{l}Q=2R_{b}L_{S2}\\ D_{S}\geq R_{a}+R_{b}\end{array} \right. \tag{6}\]  
where, \(Q\) is the distance between the two anchoring ships; \(D_{S}\) is the distance between the anchoring ships; \(R_{b}\) is the anchorage circle radius of ship B; and \(L_{S2}\) represents the length of ship B. Figure 5 shows the extreme motion dynamics of two anchoring ships which were not in the synchronous motion. At this time, the two anchoring ships are in opposite directions and their stems are opposite, which is the most dangerous state. Once a ship's anchor drags out of control, there is a danger of collision. Therefore, the two anchoring ships will set a certain safety margin while determining the anchorage circle radius, which can be expressed as Equation (7), as shown in Figure 5. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{1}{c}{**Standards**} & \multicolumn{1}{c}{**Conditions**} & \multicolumn{1}{c}{**Anchoring**} & \multicolumn{1}{c}{**The Length of Outgoing**} \\ \cline{3-4} \multicolumn{1}{c}{} & & **Conditions** & **Chain** \\ \hline \multirow{2}{*}{Chinese (General ship)} & & Wind Force \(\leq\) 7 (Beaufort) & 3H + 90 \\ \cline{2-4}  & & Wind Force \(>\) 7 (Beaufort) & 4H + 145 \\ \hline \multirow{4}{*}{Japanese, British} & Offshore/Onshore Waiting or Loading and Unloading & Good Anchor Gripping Conditions & 6H \\ \cline{2-4}  & Cargoes & Bad Anchor Gripping Conditions & 6H + 30 \\ \cline{2-4}  & Anchoring in Storms & Wind Speed 20 m/s & 3H + 90 \\ \cline{2-4}  & & Wind Speed 30 m/s & 4H + 145 \\ \hline \multirow{2}{*}{Dindar Oz} & & & \(R=\sqrt{\left(25\sqrt{H}\right)^{2}-H^{2}}\) \\ \cline{2-4}  & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Length of outgoing chain proposed by different countries. \[\left\{ \begin{array}{l}R_{AS}=R_{a}+\triangle R_{a}\\ R_{BS}=R_{b}+\triangle R_{b}\\ Q^{\prime}=2R_{b}+\triangle R_{a}+\triangle R_{b}-L_{S2}\\ D^{\prime}_{S}=R_{AS}+R_{BS}\end{array} \right. \tag{7}\]  
where, \(R_{AS}\) is the safety anchorage circle radius of ship A; \(R_{a}\) is the anchorage circle radius of ship A. In navigation practice, it is difficult for ship B to gain the anchoring radius \(R_{a}\) of ship A, because the anchoring ship A reciprocates irregularly in the radial direction within the anchorage circle. Additionally, the reasonable distance can only be determined according to the data from ship B. The distance \(Q\) shown in Equation (7) is normally used to determine the radius of the anchor position. Considering motion features of the anchoring ship, the distance \(Q\) is a variable. The anchorage circle radius based on distance \(Q\) may result in a small value, thus leading to a situation similar to the presence of obstacles in the anchorage circle in Figure 4. In order to avoid it, a larger anchorage circle radius would be chosen, which objectively wastes the anchorage space. Moreover, it is safe and reliable to use the safety distance \(D^{\prime}_{S}\) and \(Q^{\prime}\) in moving area detection if there is an accurate anchor position. In addition, considering ships passing through the anchorage, Equation (6) adds a parameter \(\tau\) as a safety margin in the anchorage circle radius model of Equation (5). Therefore, the safety distance model of two anchoring ships can be further derived, as shown in Equation (8). \[\left\{ \begin{array}{l}R_{AS}=\sigma_{a}(R_{a}+\tau_{a}B_{a})\\ R_{BS}=\sigma_{b}(R_{b}+\tau_{b}B_{b})\\ D^{\prime}_{S}=R_{AS}+R_{BS}\end{array}