3.1 Calculation of DCPA and TCPA

Firstly, the collision risk between ships in the encounter cluster is calculated. This paper uses the method of collision risk based on AIS data. Since the AIS data are received intermittently and the interval time is determined according to the ship type and ship speed, the AIS data of all ships in the study area require interpolation calculation. In this paper, the cubic spline interpolation is used to process AIS data to obtain AIS data of ships at the same time (Kouibia and Pasadas, Reference Kouibia and Pasadas2012). At the same time, there are many small ships not equiped with AIS in the restricted waters; in this situation, the ship position data should be obtained by radar, and our method can also process the ship position, speed and course from the radar data.

After obtaining accurate AIS data, the DCPA and TCPA between ships are calculated. For any pair of ships, one of them is identified as OS, and the ship parameters are $({\textrm{lon}{\textrm{g}_o},\textrm{la}{\textrm{t}_o},\textrm{so}{\textrm{g}_o},\textrm{co}{\textrm{g}_o}} )$, and the TS parameter is $\; ({\textrm{lon}{\textrm{g}_t},\textrm{la}{\textrm{t}_t},\textrm{so}{\textrm{g}_t},\textrm{co}{\textrm{g}_t}} )$. The DCPA and TCPA calculation method between ships in the encounter cluster is as follows (Figure 2):

Figure 2. The relative position of ships

The relative distance between the two ships is D_ot. Because the position information in AIS data is latitude and longitude, in order to obtain a more accurate distance between the two ships, this paper uses the Mercator method to calculate the real distance. The Mercator method is the longitude-difference calculation method, based on the characteristics of Mercator projection with equal angle and loxodrome as a straight line. The specific formulas for calculating the distance between the two ships are

(3.1)\begin{align}\textrm{MP} & = 7915 {\cdot}70447\lg \left( {\textrm{tan}\left( {\frac{\pi }{4} + \frac{{\textrm{lat}}}{2}} \right){{\left( {\frac{{1 - e\,\textrm{sin}({\textrm{lat}} )}}{{1 + e\,\textrm{sin}({\textrm{lat}} )}}} \right)}^{e/2}}} \right)\end{align}

(3.2)\begin{align}\textrm{DMP} & = \textrm{M}{\textrm{P}_t} - \textrm{M}{\textrm{P}_o}\end{align}

(3.3)\begin{align}C & = \arctan \left( {\frac{{D\textrm{long}}}{{\textrm{DMP}}}} \right)\end{align}

(3.4)\begin{align}S & = \textrm{Dlat} {\cdot}\textrm{sec}(C )\end{align}

Where MP is meridianal part, lat is latitude, e is the eccentricity of Earth and e in index e/2 is Euler number, e = 2⋅7182828459045 and DMP is the difference of meridianal parts between the two ships. The Dlong is the longitude difference between the two ships, Dlat is the latitude difference between the two ships and S is the distance between the two ships. The unit is n mile.

True azimuth of the target ship θ:

(3.5)\begin{gather}\left\{ \begin{array}{@{}l} \Delta x = \textrm{lon}{\textrm{g}_t} - \textrm{lon}{\textrm{g}_o}\\ \Delta y = \textrm{la}{\textrm{t}_t} - \textrm{la}{\textrm{t}_o} \end{array} \right. \end{gather}

(3.6)\begin{gather}\theta = \left\{ {\begin{array}{*{20}{l}} {{{\tan }^{ - 1}}\left( {\dfrac{{\Delta x}}{{\Delta y}}} \right)}& {\Delta x \ge 0,\Delta y > 0}\\ {{{\tan }^{ - 1}}\left( {\dfrac{{\Delta x}}{{\Delta y}}} \right) + \pi }& {\Delta x \ge 0,\Delta y < 0}\\ {{{\tan }^{ - 1}}\left( {\dfrac{{\Delta x}}{{\Delta y}}} \right) + \pi }& {\Delta x < 0,\Delta y < 0}\\ {{{\tan }^{ - 1}}\left( {\dfrac{{\Delta x}}{{\Delta y}}} \right) + 2\pi }& {\Delta x < 0,\Delta y > 0}\\ {\dfrac{\pi }{2}}& {\Delta x > 0,\Delta y = 0}\\ {\dfrac{{3\pi }}{2}}& {\Delta x < 0,\Delta y = 0} \end{array}} \right.\end{gather}

The speed component of the OS on the coordinate axis:

(3.7)\begin{equation}\left\{ \begin{array}{l} {v_{ox}} = {v_o} {\cdot}\sin ({\textrm{co}{\textrm{g}_o}} )\\ {v_{oy}} = {v_o} {\cdot}\cos ({\textrm{co}{\textrm{g}_o}} )\end{array} \right.\end{equation}

The speed component of the TS on the coordinate axis:

(3.8)\begin{equation}\left\{ \begin{array}{l} {v_{tx}} = {v_t} {\cdot}\sin ({\textrm{cog}{}_t} )\\ {v_{ty}} = {v_t} {\cdot}\cos ({\textrm{cog}{}_t} )\end{array} \right.\end{equation}

The relative velocity (${v_r}$) of the two ships and their components on the two axes:

(3.9)\begin{gather}\left\{ \begin{array}{l} {v_{rx}} = {v_{tx}} - {v_{ox}}\\ {v_{ry}} = {v_{ty}} - {v_{oy}} \end{array} \right.\end{gather}

(3.10)\begin{gather}{v_r} = \sqrt {{{({{v_{rx}}} )}^2} + {{({{v_{ry}}} )}^2}}\end{gather}

Relative course of two ships (${\varphi _r}$):

(3.11)\begin{equation}{\varphi _r} = \left\{ {\begin{array}{*{20}{l}} {{{\tan }^{ - 1}}\left( {\dfrac{{{v_{rx}}}}{{{v_{ry}}}}} \right)}& {{v_{rx}} \ge 0,\,{v_{ry}} > 0}\\ {{{\tan }^{ - 1}}\left( {\dfrac{{{v_{rx}}}}{{{v_{ry}}}}} \right) + \pi }& {{v_{rx}} \ge 0,\,{v_{ry}} < 0}\\ {{{\tan }^{ - 1}}\left( {\dfrac{{{v_{rx}}}}{{{v_{ry}}}}} \right) + \pi }& {{v_{rx}} < 0,\,{v_{ry}} < 0}\\ {{{\tan }^{ - 1}}\left( {\dfrac{{{v_{rx}}}}{{{v_{ry}}}}} \right) + 2\pi }& {{v_{rx}} < 0,\,{v_{ry}} > 0}\\ {\dfrac{\pi }{2}}& {{v_{rx}} > 0,\,{v_{ry}} = 0}\\ {\dfrac{{3\pi }}{2}}& {{v_{rx}} < 0,\,{v_{ry}} = 0} \end{array}} \right.\end{equation}

The DCPA can be calculated as

(3.12)\begin{equation}\textrm{DCPA} = {D_{ot}}\sin ({{\varphi_r} - \theta - \pi } )\end{equation}

If the TS passes from the starboard side of the OS when it reaches the distance to the closest point of approach if the DCPA is negative, the TS is in the head direction of the ship, but if the DCPA is positive, the TS is astern of the OS. If the TS passes from the port side of the OS when it reaches the distance to the closest point of approach if the DCPA is negative, the TS is in the stern direction of the ship, and if the DCPA is positive, the TS is in the bow direction of the ship.

The TCPA is calculated as

(3.13)\begin{equation}\textrm{TCPA} = \frac{{{D_{ot}}\cos ({{\varphi_r} - \theta - \pi } )}}{{{v_r}}}\end{equation}

If TCPA > 0, the two ships have not yet entered the closest point of approach point. If TCPA < 0, the two ships have passed the closest point of approach.

3.2 Modeling of spatial and temporal ship collision risk

There are many methods to study ship collision risk. For example, Mou et al. (Reference Mou, Van der Tak and Ligteringen2010) incorporated basic risks into the risk measurement of the studied waters and expressed the relationship between DCPA and TCPA and risks in a negative exponential form. Zhen et al. (Reference Zhen, Riveiro and Jin2017) considered the impact of DCPA and TCPA on risk and used the negative index method to calculate the collision risk. In addition to collision risk related to DCPA and TCPA. In this paper, we incorporate the safe distance (D) and time (T), to the traditional negative exponential function from the aspect of collision avoidance, to quantify the relationship between collision risk and DCPA and TCPA.

The spatial collision risk caused by DCPA can be expressed as

(3.14)\begin{equation}\textrm{CR}{\textrm{I}_{\textrm{DCPA}}} = {k_1}{e^{{\alpha _1} {\cdot}\frac{{\textrm{DCPA}}}{D}}}\end{equation}

where D is the safe distance between two ships, and ${\alpha _1}$ and ${k_1}$ are the regulating coefficients.

The temporal collision risk caused by TCPA can be expressed as

(3.15)\begin{equation}\textrm{CR}{\textrm{I}_{\textrm{TCPA}}} = {k_2}{e^{{\alpha _2} {\cdot}\frac{{\textrm{TCPA}}}{T}}}\end{equation}

where T is the safest time of the two ships, the shortest time for the two ships to avoid collision by manipulating the ship, and ${\alpha _2}$ and ${k_2}$ are the adjustment coefficients.

3.3 Collision risk modeling of ship aggregate density in clusters of encounter ships

When a ship encounters another ship at sea, it is often necessary to adjust the navigation state of the ship in order to avoid the urgent situation of a collision. Due to the large inertia of a ship, taking speed change for collision avoidance will cause more waste of resources, so most collision avoidance operations are to change the ship course. The situation of surrounding ships should be considered when changing course. The smaller the distance between the surrounding ship and the OS is, the more factors must be considered when the OS changes its course. According to the International Regulation for the Prevention of Collision at Sea (COLREGS), allowing give-away ships to turn starboard when encountering to avoid collisions in a crossing situation, it can be concluded that under the same circumstances, the ships have a greater risk of collision on the right side than on the left side of the ship. The collision risk of the ship's aggregation coefficient (AC) is the aggregation risk of OS by a single ship in the CES. The AD-based collision risk is the result of the collision risk synthesis of the AC of the ship and the surrounding ships, which reflects the density of the ships around the ship. In this paper, the smooth circular ship domain by Davis et al. (Reference Davis, Dove and Stockel1980) is used to calculate the ship domain overlap index, which is used as the influencing factor of the aggregation coefficient. Using the ship domain overlap index as an influencing factor not only represents the influence of the distance between ships but also reflects that the risk of the right side of own ship is higher than that of the ship on the left side of the ship under the same conditions. The center of the ship domain is located on the upper-right of the ship, and the specific parameters are shown in Figure 3.

Figure 3. The ship domain used in this paper

The ship density refers to the number of ships in a certain instantaneous unit area of water. This can also reflect the ship aggregation in the study area, but the effect of other ships on the ship can influence the evasive actions. As shown in Figure 4, although the density of ships in the area is the same, the aggregation of each ship in the area is not the same, so the density of ships cannot effectively represent the risk of the location of the ship in the unit water area.

Figure 4. Diagram of ship density

It is assumed that the radius of the ship domain is ${r_o}$, the ship domain of the target ship is ${r_t}$, and the distance between the virtual positions of the two ships is ${D_{ot}}$, and the ship domain overlap index can be expressed as (Liu et al., Reference Liu, Wu and Zheng2019)

(3.16)\begin{equation}\textrm{SDO}{\textrm{I}_{ot}} = \frac{{{D^{\prime}}_{ot}}}{{{r_o} + {r_t}}}\end{equation}

The AC essentially reflects the influence of other ship's state on the ship's manipulation. The smaller the SDOI, the greater the impact of the other ships on the own ship, so the SDOI of the own ship and the other ships are negatively correlated with the collision risk of the ship's AC. Combined with spatial collision risk and temporal collision risk, this paper also uses the negative exponential to express the relationship between SDOI and ship AC collision risk. The details are

(17)\begin{equation}\textrm{CR}{\textrm{I}_{\textrm{A}{\textrm{C}_{ot}}}} = {k_3}{e^{{\alpha _3}\textrm{SDO}{\textrm{I}_{ot}}}}\end{equation}

According to Davis's research, the scope of the threat to the officers is further than the ship domain boundary, and a super-domain is determined. The scope of the super-domain is that the officers start to take action to avoid the distance from other ships when an urgent situation occurs, also known as the ‘Arena’. Since the AC is the influence on the ship maneuvering, it is necessary to calculate the ship aggregation within the dynamic range. Figure 5 shows the Arena model adopted in this paper and the ships involved in calculating the AC around the ship.

Figure 5. The ship domain and Arena

The dashed line circle represents the Arena and the solid line circle represents the ship domain. The Arena is tangent to the ship domain and the tangent points are collinear with two centers. In Figure 5, only ship₁ and ship₂ are required to calculate the AC of the ship. Ship₃ is not in range of there Arena, and here we assume there is no effect on the OS's manipulation.

In Figure 6, the OS is indicated in center, and the relative positions of the other ships are shown around in four directions. Assuming that the length of the own ship and other ship are both 100 m, and Figures 7–10 show the AC-based collision risk around OS based on Equations (3.16) and (3.17). These figures show that at different encounter angles, even if the distance from the OS is the same, the collision risk of the AC is different.

Figure 6. Diagram of relative course between OS and other ships

Figure 7. The relative course of 0°

Figure 8. The relative course of 90°

Figure 9. The relative course of 180°

Figure 10. The relative course of 270°

When there is only one ship within the OS Arena, the aggregation coefficient is equal to the AD. When there are many ships within the OS Arena, it is necessary to synthesise the aggregation coefficient of all ship pairs to obtain the comprehensive AD. To reduce the interference of human factors on the AD calculation, and fully reflect that the closer the ship is, the greater the risk of AD is. The calculation formula of AD-based collision risk from all the influences in the CES is

(3.18)\begin{equation}\textrm{ADR}{\textrm{I}_i} = 1 - \prod\limits_{j \ne i} {({1 - \textrm{CR}{\textrm{I}_{\textrm{A}{\textrm{C}_{ij}}}}} )}\end{equation}

For example, when the AC of three ships in the dynamic range of the ship is 0⋅6, 0⋅4 and 0⋅2, respectively, the calculated AD is 0⋅808. In principle, there should be no other ships in the Arena range of the OS, so the corresponding aggregation density of the three ships will be large. This formula can fully reflect the influences of AD in the CES.

3.4 Regional collision risk modeling of encountering ship clusters

The AD of CES can reflect the influence of the surrounding ship on the ship's motion. When there is no ship within the OS Arena, the ship's AD is 0, and the ship's freedom is high. The traditional density calculation method cannot fully reflect the freedom of navigation of the ship. The method of combining ship AD, DCPA and TCPA to identify regional ship collision risk has advantages over the existing method.

All ships sailing at sea are a dynamic system, and the ships in the system interact with each other. Taking ship A as the research object, the collision risk of ship A and ship B cannot only consider the mutual influence between ship A and ship B but also consider the influence of other ships in the system on ship A. The farther the distance is, the smaller the influence is. This paper is based on the Arena of ship A, and the influence of ships outside Arena on ship A is negligible. This effect is expressed by the ship's AD. The ADRI_i is the combination of AD-based risk. The ship collision risk is determined by spatial collision risk, temporal collision risk, and AD-based collision risk. Space and temporal collision risk belong to the influence between ship pairs, and AD-based collision risk belongs to the influences of all the ships in CES on OS. Collision risk can be identified by the linear combination of the three aspects of collision risk, then the specific formula of collision risk of ship_j and ship_i in CES can be expressed as

(3.19)\begin{equation}\textrm{R}{\textrm{I}_{ij}} = {\omega _1}\textrm{CR}{\textrm{I}_{\textrm{DCPA}}} + {\omega _2}\textrm{CR}{\textrm{I}_{\textrm{TCPA}}} + {\omega _3}\textrm{ADR}{\textrm{I}_\textrm{i}}\end{equation}

where the ${\omega _1}$, ${\omega _2}$ and ${\omega _3}$ are the weights of spatial collision risk, temporal collision risk and AD-based collision risk, respectively.

After calculating the collision risk, the researchers studied it from a deeper perspective. Zhang et al. (Reference Zhang, Kopca, Tang, Ma and Wang2017) used expert judgement to sort ship collision risks, and experts further reviewed these risks. There are few studies on the interaction between ships in the system to determine the collision risk of a single ship. Liu et al. (Reference Liu, Wu and Zheng2019) applied the Shapley value method in game theory to determine the weight of each collision risk, and finally accumulated the collision risk of a single ship. Feng et al. (Reference Feng, Zhang and Qi2018) constructed the risk matrix and calculated the mean value of the risk matrix as the collision risk of a single ship. These methods have some limitations to obtain the ship collision risk. For example, the risk of collision between OS and several other ships is 0⋅5, so whatever method is used to set the weight, the final risk is 0⋅5. According to common sense, if many ships influence each other at this time, the risk of the ship must be greater than these risks. To overcome this limitation, this paper proposes a method to calculate the total risk of ship collision without weight setting. According to the Equation (3.19), the value range of RI_ij is 0 ~ 1. So the specific formula is

(3.20)\begin{equation}\textrm{R}{\textrm{I}_i} = 1 - \prod\limits_{j \ne i} {({1 - \textrm{R}{\textrm{I}_{ij}}} )}\end{equation}

Finally, the collision risk of each ship is identified from the CES, and the obtained collision risk is displayed on the map in the form of a heatmap. This not only reminds the crew of the dangerous degree of their environment, but also makes judgements on areas with high collision risk VTS operators, and reminds the ship to pay attention to navigation safety. Figure 11 shows the whole steps for identifying regional collision risks considering the AD.

Figure 11. The regional collision risk identification process

4.1 Parameter selection of DBSCAN algorithm

Firstly, the DBSCAN algorithm is used to cluster the ship position points. The MinPts of DBSCAN algorithm must be equal or more than two, because the ship encounter are formed by at least two ships. Secondly, clustering results cannot produce too much noise, which will affect data integrity and reduce risk assessment. Finally, the ships in each cluster cannot be too numerous or they will lose the meaning of clustering. To get better clustering results, we have carried out many experiments on setting two parameters of MinPts and Eps, which are shown in Figure 13(a). From the clustering experiments, we know that the larger the Eps parameters are and the smaller the Minpts parameters, the smaller the amount of noise data will be.

Figure 13. (a) Relationships between Eps, minPts and noise quantitative, (b) relationship between Eps, minPts and cluster number

According to Figure 13(b), by setting the same Eps, the smaller the MinPts are, the more clusters are obtained. Too many clusters will seriously segment the study area, resulting in large collision risk error. Considering the amount of generated noises and the number of clusters, MinPts is set to 3, and Eps is set to 1⋅2 nm. There are nine CES within the Xiamen Bay waters, as shown in Figure 14.

Figure 14. The DBSCAN clustering results in Xiamen waters

4.2 Determine the model parameters

The parameters are set from the empirical cases and the expert's experience. Li (Reference Li2019) suggested that the influences from other ships away from 1 n mile can be ignored for 20 m ships, and other ships away from 3 n miles can be ignored for 400 m ships. We use linear interpolation to obtain the negligible DCPA for ships of different lengths. In Table 1, the term ‘4L/1850’ indicates that the DCPA is the size of the ship's domain. When DCPA is greater than the ignored distance, the risk calculated by DCPA can be ignored. When the DCPA is equal to the size of the ship doamin, the collision risk value is large. The TCPA and DCPA based collision risk calculation are are designed by Zhen et al. (Reference Zhen, Riveiro and Jin2017), when TCPA is greater than 30 min, the calculated temporal collision risk is less than 0⋅1. When TCPA is less than 2 min, the calculated temporal collision risk is greater than 0⋅8. When SDOI is 0⋅1, the AD is about 0⋅9. When SDOI is 1, the AD is about 0⋅3. Figure 15 shows the effects of DCPA, TCPA and density on CRI.

Figure 15. (a) The effect of DCPA and TCPA on CRI, (b) The effect of DCPA and density on CRI, (c) The effect of TCPA and density on CRI

Table 1. Parameters calculation of CRI_DCPA

4.3 Calculation of ship collision risk

The safety distance in Equation (3.14) is four times the ship length, and the safety time in Equation (3.5) is 10 min. The weight of spatial collision risk, time collision risk and AD collision risk are 1/3. To verify the effectiveness of this method, the 9th clusters of encounter ships are selected for analysis. There are five ships in this CES, and the specific parameters of these five ships are shown in Table 2.

Table 2. Specific parameters of ships

The DCPA and TCPA matrix are obtained as follows:

(4.21)\begin{equation}\left[ {\begin{matrix} 0& { - 0 {\cdot} 025}& { - 0 {\cdot} 407}& { - 0 {\cdot} 927}& {0 {\cdot} 994}\\ { - 0 {\cdot} 025}& 0& {0 {\cdot} 130}& { - 0 {\cdot} 685}& {0 {\cdot} 656}\\ { - 0 {\cdot} 407}& {0 {\cdot} 130}& 0& { - 0 {\cdot} 496}& {0 {\cdot} 642}\\ { - 0 {\cdot} 927}& { - 0 {\cdot} 685}& { - 0 {\cdot} 496}& 0& { - 0 {\cdot} 057}\\ {0 {\cdot} 994}& {0 {\cdot} 656}& {0 {\cdot} 642}& { - 0 {\cdot} 057}& 0 \end{matrix}} \right]\end{equation}

(4.22)\begin{equation}\left[ {\begin{matrix} 0& { - 50 {\cdot} 2}& {2 {\cdot} 4}& { - 0 {\cdot} 61}& { - 1 {\cdot} 7}\\ { - 5 {\cdot} 2}& 0& {1 {\cdot} 5}& {0 {\cdot} 15}& {0 {\cdot} 21}\\ {2 {\cdot} 4}& {1 {\cdot} 5}& 0& { - 4 {\cdot} 2}& { - 0 {\cdot} 002}\\ { - 6 {\cdot} 1}& {0 {\cdot} 15}& { - 0 {\cdot} 42}& 0& {0 {\cdot} 80}\\ { - 1 {\cdot} 7}& { - 0 {\cdot} 21}& { - 0 {\cdot} 002}& {0 {\cdot} 80}& 0 \end{matrix}} \right]\end{equation}

According to Equations (3.16), (3.17), and (3.18), the collision risk of vessel AD can be obtained as

(4.23)\begin{equation}\left[ {\begin{matrix} {\textrm{0} {\cdot}\textrm{381}}& {\textrm{0} {\cdot}\textrm{522}}& \textrm{0}& \textrm{0}& \textrm{0} \end{matrix}} \right]\end{equation}

According to Equation (3.19), the collision risk matrix is

(4.24)\begin{equation}\left[ {\begin{matrix} 0& 0& 0& 0& 0\\ 0& 0& {0 {\cdot}79}& 0& 0\\ 0& {0 {\cdot}602}& 0& 0& 0\\ 0& 0& 0& 0& {0 {\cdot}709}\\ 0& 0& 0& {0 {\cdot}695}& 0 \end{matrix}} \right]\end{equation}

According to Equation (3.10), the collision risk of a single ship in CES₅ is

(4.25)\begin{equation}\left[ {\begin{matrix} 0& {0 {\cdot}79}& {0 {\cdot}602}& {0 {\cdot}709}& {0 {\cdot}695} \end{matrix}} \right]\end{equation}

There are five ships displayed on the map as shown in Figure 16, the arrow length represents the relative speed of the ship. The collision risk of each ship has been influenced by other 4 ships in the 9th CES, the OS denotes each ship when calculate the collision risk from other ships in the same CES.

Figure 16. The spatial distribution of ships in 9th cluster of encounter ships.

Due to the different lengths and relative positions of ships, the obtained risk matrix is asymmetric. Ship₁ and ship₂ have passed the CPA, so the collision risk of the two ships is 0. The ship₁ and ship₃ will encounter a distance of 0⋅407 n miles in the later time, which is not within the Arena range of ship₁ and ship₃, so the collision risk of ship₃ and ship₁ is 0. ship₁ and ship₂ have other ships within their Arena range. These ships will have impacts on the collision avoidance of the ships, and the AD of these ships must be considered. Based on the model proposed, the risks of ship₁ and ship₂ due to the aggregation of surrounding ships are calculated to be 0⋅381 and 0⋅522, respectively. The situation of ship₄ and ship₅ belong to the crossings situation, and the nearest encounter distance is within the dynamic range of the ship, but there is no ship within the dynamic range, so only spatial collision risk and temporal collision risk are considered. The collision risks of five ships in CES 9th are calculated as follows: 0, 0⋅79, 0⋅602, 0⋅709 and 0⋅695, respectively, as shown in Figure 17.

Figure 17. The regional collision risk of 9th cluster of encounter ships

After calculating the collision risk of all single ships in the CES, we depict the whole spatial distribution of regional collision risk by heatmap on the nautical chart, as shown in Figure 18, the colours blue to red represents the risk from low to high. The depth of the colour depends on the magnification of the map: the higher the magnification, the more blurred the risk display, and the smaller the magnification, the more accurate the risk display. This is done so that maritime traffic management officers can find areas of higher risk quickly based on our proposed method in this paper.

Figure 18. The distribution of regional ship collision risk within Xiamen Bay