## 4 Re-Designing of the Appropriate Fairway Width
### Assessment Overview
\[ES_{L}=\sum_{\varphi=-90^{\circ}}^{+90^{\circ}}S_{L} \tag{5}\]  
\[S_{L}=\alpha\times(R/V)+\beta \tag{6}\]  
where  
\(S_{L}\): Risk of obstacles;  
\(R\): Distance to obstacles;  
\(V\): Speed of the own ship;  
\(\alpha,\beta\): Coefficient determined by natural conditions;  
\(\alpha=-0.00092\times\log 10(GT)+0.0099\)  
\(\beta=-3.82\). The calculation method of \(ES_{S}\) is as Equations (7) and (8) [37]. \[ES_{S}=\sum_{\varphi=-90^{\circ}}^{+90^{\circ}}S_{S} \tag{7}\]  
\[S_{S}=\alpha\times(R/V\times V/L_{m})+\beta=\alpha\times(R/L_{m})+\beta \tag{8}\]  
where  
\(\alpha=0.00192\times L_{m}\):  
In case the condition of encounter with the target ship is:  
Crossing from starboard, \(\beta=-0.65\times\ln Lm-2.07\). If crossed from portside, \(\beta=-0.65\times\ln Lm-2.35\). If meeting from the bow, \(\beta=-0.65\times\ln Lm-2.07\). If the ship is overpassed from the stern, \(\beta=-0.65\times\ln Lm-0.85\). \(S_{S}\): Risk concerning the target ship;  
\(R\): Relative distance between own ship and target ships;  
\(V\): Relative speed between own ship and target ships;  
\(Lm\): Average length of the ship and target ships. Equation (9) is a formula for the comprehensive evaluation of \(ES_{L}\) and \(ES_{S}\)[37]. \[ES_{A}=\sum_{\varphi=-90^{\circ}}^{+90^{\circ}}\max\{S_{L},S_{S}\} \tag{9}\]  
2. IWRAP Grounding Probability  
IALA Waterway Risk Assessment Program (IWRAP Mk II)--a maritime risk assessment tool--quantifies, predicts, and evaluates risks related to maritime traffic. IWRAP Mk II is used as a useful modeling tool for maritime risk assessment and estimates the frequency of collisions and grounding with ships, based on traffic volume, traffic distribution, and seabed topography information [41]. In IWRAP, the types of situations in which grounding may occur are the following:  
1. A vessel navigating at a normal speed on a normal fairway caused by human error,  
2. a vessel that fails to change course at a changing point near an obstacle,  
3. a vessel that takes an evasive action near an obstacle. The ships on grounding accident are classified into types  
1,
2, and  
3, including the types of accidents,  
4. loss of propulsion, etc., and the formula applied according to the type of situation is also different. In this study, only categories 1 and 2 are reflected in the evaluation to obtain the probability of grounding in the ship's operation and changing direction under normal circumstances [42; 43]. The method of calculating the grounding probability for categories 1 and 2 was as shown in Equations (10) and (11):  
\[\mathrm{N_{I}}=\sum_{Ship\ Class,\ i}P_{c,i}Q_{i}\int_{z_{min}}^{z_{max}}f_{t} (z)dz \tag{10}\]  
\[\mathrm{N_{II}}=\sum_{Ship\ Class,\ i}P_{c,i}Q_{i}\exp(-d/a_{t})\int_{z_{min}}^ {z_{max}}f_{t}(z)dz \tag{11}\]  
where \begin{tabular}{l l} \(a_{t}\) & Average distance between position checks by the navigator; \\ \(d\) & Distance from the obstacle to the bend in the navigation route varying with the lateral \\ \(d\) & position, s, of the ship; \\ \(i\) & Index for ship class, categorized after vessel type and dead weight or length; \\ \(f_{t}(z)\) & Probability density function for the ship traffic; \\ \(\mathrm{N_{I}}\) & Expected number of category I grounding events per year; \\ \(\mathrm{N_{II}}\) & Expected number of category II grounding events per year; \\ \(P_{c,i}\) & Causation probability, i.e., ratio between ships grounding and ships on a \\  & grounding course; \\ \(Q_{i}\) & Number of ships in class i passing a cross section of the route per year; \\ \(z\) & Coordinate in the direction perpendicular to the route; \\ \(z_{min},z_{max}\) & Transverse coordinates for an obstacle. \\ \end{tabular}   
In this study, based on the GICOMS data obtained for 7 days, the frequency of grounding accidents with obstacles near each target fairway was estimated reflecting the maritime traffic flow. 3.