2.1. Defining the composition of the NVCRO mathematical model

Maritime traffic conditions are recorded in AIS data, which includes time, longitude, latitude, speed, heading and the size and type of vessels (Figure 2). The mathematical model for NVCRO has a certain importance for the definition of the collision risk matrix and network which reflects the possibility and severity of the risk of collision between two vessels. Thus, in this paper, we take possible relative striking location and safety domain of the ships into consideration. The factors that affect the analysis of NVCRO are analysed as follows.

Figure 2.

The basic factors of vessel collision risk and the construction of the NVCRO.

2.1.1. Safety distance between encountered vessels

This is the closest distance between the ships' safety domain boundaries. Previous studies have studied the safety domain boundary. Szlapczynski and Szlapczynska (Reference Szlapczynski and Szlapczynska2017) discussed a number of ship domains and considered their difference not only in size or shape but also in their purposes and definition. Wang et al. (Reference Wang, Meng, Xu and Wang2009) mentioned several approaches to measure the size of the safety domain, including ellipses, circles and polygons. This study investigates marine traffic in the Baltic Sea, an elliptical region mentioned by Yahei and Reijiro (Reference Yahei and Reijiro1971), due to the huge number of ships operating in this area with a wide variety of applications. Equations (1)–(3) are calculated based on the elliptical shape of the ship domain. If other shapes of ship domain are adopted, the following equations and results would need to be adapted accordingly.

The safety domain is usually proportional to ship size, which is defined as an area around a ship to be kept clear of other ships in order to avoid collisions (Wang and Chin, Reference Wang and Chin2016; Wang et al., Reference Wang, Meng, Xu and Wang2009). As shown in Figure 3, the safety distance l _ij is the relative distance between the encountered ships' closest safety domain boundaries, which can be calculated by the following equations:

(1)$$\alpha_i =arccos\left(\displaystyle{{y_j-y_i}\over{\sqrt{\lpar x_j-x_i\rpar^2+\lpar y_j-y_i\rpar^2}}}\right)-\theta_i $$

(2)$$\alpha_j = \pi-\alpha_i-\theta_{ij} $$

(3)$$l_i = \sqrt{\displaystyle{{1+\tan^2\alpha_i}\over{\displaystyle{{1}\over{\lpar 1{\cdot}6L_i\rpar ^2}}+\displaystyle{{\tan^2\alpha_i}\over{\lpar 4L_i\rpar ^2}}}}} $$

(4)$$l_{ij} = Dist_{ij}-l_i-l_j $$

where α _i is the angle between the sailing direction of ship i and the linking line between ship i and ship j. θ _i is the angle between the sailing direction of ship i and longitude. (x _i, y _i) and (x _j, y _j) are the latitude and longitude coordinates of ship i and ship j, and L_i and L_j are the lengths of ship i and ship j. L_ij is the safe distance between ship i and ship j, and l _i is the distance between ship i and ship j. The distance Dist _ij between ship i and ship j is in the same direction of l _i and l _j, and furthermore, the calculation of these three distances are aligned with the direction of the link between geometric centres of both encountered ships i and j.

Figure 3.

The safety distance and length of ship I and ship j.

2.1.3. Relative orientation of encountered vessels

In a mathematical model, the angle represents the relative heading of the bow of the two ships. Zhang et al. (Reference Zhang, Goerlandt, Montewka and Kujala2015b) defined the term ‘phase’ as positive or negative angles to imply that ships are converging (+ve) or diverging (-ve) from each other. This study uses the concept of Minimum Distance To Collision (MDTC) (Montewka et al., Reference Montewka, Hinz, Kujala and Matusiak2010), which is defined as the minimum safety distance to avoid a collision at different angles of intersection between vessels. Zhang et al. (Reference Zhang, Goerlandt, Kujala and Wang2016) fitted a MDTC curve, which allows a better distinction of the risk levels at various encounter angles. Under the different angles of intersection between ships, data from Montewka et al. (Reference Montewka, Hinz, Kujala and Matusiak2010) are adopted to describe the conflict severity. A larger MDTC indicates that a greater distance is necessary for a ship to manoeuvre to avoid collision. Zhang et al. (Reference Zhang, Goerlandt, Kujala and Wang2016) introduced C _MDTC to measure the relationship between conflict severity and the intersection angle, which is formulated based on a Fourier series, as shown in Equation (5).

(5)$$C_{MDTC} \sim \sum_{k=1}^n {m_k \sin \lpar k\cdot \theta _{ij} \rpar } $$

where n is the expansion number of the Fourier series, θ _ij is the intersecting angle of the two ships, and the Fourier expansion coefficient is m _k. Equation (5) describes a sine series; the angle of θ _ij in the range of (−π, 0) results in the two vessels diverging, whereas a positive value represents ships approaching.

2.1.5. Possible relative striking location based on interval analysis

Based on interval analysis, the collision type of the target ship (red in Figure 4) is divided into five types: head-head collision, rear-end collision, front side-touch, back side-touch and frontal collision. Figure 4 shows these possible relative striking locations determined by interval analysis. The severity of damage from the different locations on the hull was assessed in Montewka et al. (Reference Montewka, Ehlers, Goerlandt, Hinz, Tabri and Kujala2014b). In this study, the possible relative striking location is further developed as an extension of the work of Montewka et al. (Reference Montewka, Ehlers, Goerlandt, Hinz, Tabri and Kujala2014b). Also, a possible violation of ship domain is used to replace the potential striking position of the hulls, which is easily handled during data processing and has an impact on collision avoidance decisions. To reflect the severity of the possible striking locations, the term ‘vessel rupture degree’ has been introduced, and the possibility of vessel rupture is one way to determine the degree of collision risk.

Figure 4.

Five types of collision and possible relative striking positions for the target ship.

The possible consequences of the collision play an important role in risk assessment. A previous VCRO model (Zhang et al., Reference Zhang, Goerlandt, Kujala and Wang2016) did not take these possible consequences into account. Therefore, the relationship between possible consequences and the severity of the collision risk is established in this study, which considers the NVCRO and vessel rupture degree as shown in Equation (6). The vessel rupture degree is dependent on the possible striking location for a given length ratio.

(6)$$NVCRO_{j\to i}\sim f\lpar R_i\lpar S_i\comma \; \theta_{ij}\comma \; \sigma_{ij}\rpar \rpar $$

Due to the different safety domains and the possible relative striking positions, each vessel has a different NVCRO value. In this paper, NVCRO_j→i is the NVCRO value when ship i encounters ship j, where θ _ij is the intersecting angle between two encountered ships, and σ _ij is the length ratio of the two encountered ships. The possible relative striking location was modelled as a uniform distribution (Montewka et al., Reference Montewka, Ehlers, Goerlandt, Hinz, Tabri and Kujala2014b) ranging from −0·5 to 0·5 (Figure 5).

Figure 5.

Possible relative striking location division of a vessel (Montewka et al., Reference Montewka, Ehlers, Goerlandt, Hinz, Tabri and Kujala2014b).

As shown in Figure 5, a head-head collision between ships results in the hull damage in ①, and a rear-end collision results in the hull being affected in ① and ⑤, respectively. Both are designated as scenario (S₁). A back side-touch collision of the hull is ② or ⑧ (S₂). Frontal collisions can occur in ③ or ⑦ (S₃). The site of collision where the front of the hull was subjected to a side-touch is ④ or ⑥ (S₄). Taking into account the possibility of the collision occurrence and a uniform distribution (Montewka et al., Reference Montewka, Ehlers, Goerlandt, Hinz, Tabri and Kujala2014b), the possible collision sites are of five types, and S_i represents five types which include a head-head collision rear-end collision, front side-touch, back side-touch and frontal collision based on the relative locations as shown in Table 1.

Table 1.

Possible relative striking location divisions

Interval analysis was used to determine the possible relative striking locations, and the sailing speed of the vessels was utilised to determine the relative striking locations. The scenarios in which two ships that are close and converging are considered. Using the intersecting angle and the ship speed, the type of collision is analysed by interval analysis, which is shown in Table 2.

Table 2.

Collision of vessel position under interval analysis of speed and relative angle

Table 3 is an extension of Montewka et al. (Reference Montewka, Ehlers, Goerlandt, Hinz, Tabri and Kujala2014b). According to this research, the striking angle in the range of 45–135^° would probably cause a rupture to the hull as determined by the possible relative striking location with a given mass ratio [0 · 6, 1 · 3]. Furthermore, in this current study, ratios greater than 1·3 or less than 0·6 are simply normalised according to ship size, resulting in a series of possible damage situations. To take the difference of ship size into account during a collision, the rupture value of a smaller ship is normalised to 1, which means a more severe risk, whereas a rupture value of the larger ship is normalised to 0·5, denoting a lesser risk. The value of 0·5 is the smallest value on the curve in Montewka et al. (Reference Montewka, Ehlers, Goerlandt, Hinz, Tabri and Kujala2014b) and refers to the lowest relative available deformation energy. The smaller the rupture value, the lower the risk of collision.

Table 3.

Rupture values based on relative angle and striking location

If the relative angle of navigation approaches 0 or 180^°, a collision between two ships will be determined by the trajectory of those ships. If they are moving in the same direction and the ship at the rear is faster than the ship at the front, then a rear-end collision will occur. Table 3 shows that the collision severity of a rear-end collision is less than the loss of the hull caused by the collision.

2.1.6. The mathematical model of NVCRO

The mathematical expression that describes the NVCRO is characterised by the effects of the risk factors and the severity of the collision risk, where the relative speed, safety distance, and heading of the ships are calculated. According to the definition of VCRO proposed by Zhang et al. (Reference Zhang, Goerlandt, Kujala and Wang2016), ships at a distance greater than 6 nm are considered safe, and the expected collision risk is defined as five, indicating a very low risk. The safety domain of vessels in this study is more realistic for open, busy waters. The general navigation risk calculation for the NVCRO is described by:

(7)$$NVCRO_{j\to i} =R\cdot \displaystyle{{\vert v_{ij}\vert}\over{Dist_{ij} -l_i -l_j}}\cdot \displaystyle{{R\lpar S_i \comma \; \theta _{ij} \comma \; \sigma_{ij}\rpar }\over{L_i }}\cdot\sum_{-\infty }^{+\infty } {m_k \cdot \sin \lpar k\cdot \theta _{ij} \rpar } $$

where R is a parameter for fitting the model described by Zhang et al. (Reference Zhang, Goerlandt, Kujala and Wang2016), |v _ij| is the scalar value of ship i speed and ship j speed, and S _i represents the category of possible relative striking locations (Table 1). The risk of hull rupture in a collision between encountered vessels is presented as R_i (S_i, θ _ij, σ _ij) (Table 3) and C _MDTC as in Equation (5). If the NVCRO value is less than zero, the two-ships would diverge, and can be ignored or simply marked as zero. Additionally, the distances from ships i and j to the safety domain boundary are l _i and l _j, respectively, and Dist _ij is the relative distance between the two ships.

2.2. Developing a collision risk matrix

Based on the mathematical model of NVCRO, the probability and possible consequences of the collision risk between two ships for ship i _m are defined by Equation (8). The vessel risk matrix is constructed for each ship as follows:

(8)$$i_m = \left(\matrix{NVCRO_{i_{m+1}\to i_m} & \cdots & 0 \cr \vdots & \ddots & \vdots \cr 0 & \cdots & NVCRO_{i_{m-1}\to i_m}}\right)$$

The vessel risk matrix in Equation (8) clearly reflects the severity of the collision risk between ship i _m and other ships for a certain area. The probability values on the main diagonal reflect the relationship between the observed and surrounding ships. The multi-vessel risk matrix from the first ship to the n-th ship is:

(9)$$i_i=\left(\matrix{NVCRO_{i_{2}\to i_1} & \cdots & 0 \cr \vdots & \ddots & \vdots \cr 0 & \cdots & NVCRO_{i_{n}\to i_1}}\right)\comma \; \ldots\comma \; i_n=\left(\matrix{NVCRO_{i_{1}\to i_n} & \cdots & 0 \cr \vdots & \ddots & \vdots \cr 0 & \cdots & NVCRO_{i_{n-1}\to i_n}}\right)$$

After the construction of the risk matrix, values of the NVCRO of ship i _m and ship i _k were extracted. Then the base vector e extracts the value of $NVCRO_{i_{k}\to i_{m}}$ and the following extraction value of $NVCRO_{i_{k}\to i_{m}}$ is:

(10)$$NVCRO_{i_{m}\to i_k}=\vert i_k\cdot e_{m-k}\vert\comma \; \quad NVCRO_{i_{k}\to i_m}=\vert i_m\cdot e_{n-m-k}\vert $$

There are several possible situations of the collision risk between two ships as follows:

1. (1) The values of $NVCRO_{i_{m}\to i_{m}}$ and $NVCRO_{i_{k}\to i_{m}}$ exist with a large difference;

2. (2) The difference between the values of $NVCRO_{i_{m}\to i_{k}}$ and $NVCRO_{i_{k}\to i_{m}}$ is small;

3. (3) $NVCRO_{i_{m}\to i_{k}}$ and $NVCRO_{i_{k}\to i_{m}}$ are similar.

2.3. Establishing the regional risk network

After defining the risk matrix, the model of the regional risk network reflects the vessel sailing conditions, which is taken as an important and unavoidable element in assessing the regional vessel collision risk (Figure 6). The approach adopted that models the regional risk network is to take a ship as a node. The node is marked with a ship organisation number, and the value of the vessel collision risk between vessels is reflected in the risk matrix, from which the link strength between the ships is calculated by the severity of the collision risk.

Figure 6.

Regional risk network diagram.

Such a network can be defined by four elements: the nodes of the network with vessels, the collision risk matrix, link strength among the nodes that is shown as the network edge (Figure 6) as determined by the severity of the collision risk (Equation (11)), and the direction of network edge from a smaller value of the NVCRO to a larger one. Zhang et al. (Reference Zhang, Goerlandt, Kujala and Wang2016) considered ships more than 6 nm away as safe, where the expected collision risk value δ was defined as five to indicate a very low risk.

(11)$$NVCRO_{i_{m}\comma i_k}=\sqrt{\displaystyle{{NVCRO_{i_{m}\to i_k}^2+NVCRO_{i_{k}\to i_m}^2}\over{2}}} $$

The network illustrates the sailing conditions of vessels, especially the severity of the collision risk. Therefore, the method of square means is applied to some vessels with a high NVCRO value, indicating a high risk. Additionally, the direction of the network edge was determined by the value of the vessel collision risk, from a small value of NVCRO to a large one. After the quantitative and qualitative analysis of ships, the direction of the edge may enhance the safety awareness for the ship with a high NVCRO. Therefore, to define the direction of the edge, Equation (12) is proposed:

(12)$$I=\max\lpar NVCRO_{i_{m}\to i_k}\comma \; NVCRO_{i_{k}\to i_m}\rpar $$

The construction of a regional risk network has three steps. First, the vessels are taken as the nodes of the regional risk network that are marked with a ship organisation number. Second, the strength of the relation between the ships that represents the severity of the collision risk is used as network edges. The direction of the edge is determined by the severity of the collision risk. Third, the risk matrix, which contains the value of NVCRO, reflects the severity of any possible collision risk when a ship encounters other ships. In order to quantify the effect of the collision risk between vessels, the NVCRO is modified to increase the prediction accuracy of the collision risk between two ships. This differs from the NVCRO mentioned in Zhang et al. (Reference Zhang, Goerlandt, Kujala and Wang2016), as we show the severity of the collision risk which includes the rupture value related to the possible relative striking location for a given length ratio.

2.4. Defining the regional vessel collision risk

The specific value of the NVCRO as an element in the collision risk matrix can reflect the regional multi-vessel collision risk. Using the MVCRO, which is composed of NVCRO data, the assessment of multi-vessel collision risk was constructed to demonstrate regional multi-vessel collision risk. If the NVCRO data have an extremely high value, the sailing situations of vessels are assumed to have a high risk. For this reason, a quadratic mean was introduced, which was affected by extremely high values in a set of data as shown in Equation (13).

As demonstrated by Wen et al. (Reference Wen, Huang, Zhou, Yang, Xiao and Wu2015), density complexity is a useful method to describe the traffic complexity. Two areas of different density that contain the same number of ships may show different traffic complexity. Simulated α and λ and the parameters which depend on the type and navigation environment of a ship are replaced by the total length of the two ships multiplied by four (Wen et al., Reference Wen, Huang, Zhou, Yang, Xiao and Wu2015). In this study, Wen's length parameter setting is used in a similar way. The ellipse safety domain is used (Zhang et al., Reference Zhang, Goerlandt, Kujala and Wang2016), and the maximum value of the distance between the ships is set at four times the total length of the two ships.

(13)$$MVCRO = \sqrt{\displaystyle{{\sum_{j\comma i=1}^{n}\lpar NVCRO_{j\to i}^2+NVCRO_{i\to j}^2\rpar }\over{2\times C_n^2}}}\comma \; \quad i\neq j $$

(14)$$den_{ij} = \lambda\cdot e^{-\alpha\displaystyle{{\vert\, {\overrightarrow{Dist_{ij}}}\vert}\over{4\cdot \lpar L_i+L_j\rpar }}}\comma \; \quad i\neq j $$

In this paper, n is the number of ships within 6 nm. In order to assess the severity of multi-vessel collision risk, the MVCRO of the sailing vessels was analysed. The analysis of MVCRO is based on the risk analysis for each pair of ships and the method of permutations is used to calculate the value of $C_{n}^{2}$. The density complexity value as an element of the regional vessel collision risk can reflect the traffic conditions in the area. After completing the two methods as described by Equations (13) and (14), the assessment of regional risk (RVCRO) is analysed using Equation (15):

(15)$$RVCRO=\displaystyle{{\sum_{1\leq i< j\leq n}den_{ij}}\over{C_n^2}}+MVCRO\comma \; \quad i\neq j $$

The severity of regional vessel collision risk was constructed as RVCRO, with multi-vessel sailing risk and density complexity. Due to the definition of density complexity in Equation (14), each pair of ships and the value of $C_{n}^{2}$ were based on the method of permutations.

3.1. The encountered scenarios of multiple vessels

The number of vessels related to the density complexity and the process of navigation in open waters is important in establishing a regional risk network and an assessment of regional collision risk. Therefore, a study of the trajectories of multiple vessels within 6 nm of each other was undertaken. A full scenario of marine safety analysis in the Northern Baltic Sea is shown in Figure 7, which is the geographical distribution of the multi-vessel collision risk. The small coloured squares are computed and ranked by the RVCRO model, where each square represents an area of 6 by 6 nm. The locations of these scenarios in this section are shown in Figure 7.

Figure 7.

The Baltic Sea scenario.

The trajectories of vessels near Stockholm (area A) in the Baltic Sea are shown in Figure 8(a), commencing at 14:33 on 8 July 2011 and finishing at 14:55 on 8 July 2011. Figure 8 shows the multi-vessel tracking close to Helsinki (area B) in the Baltic Sea, with a start time of 06:17 on 8 July 2011 and a finish time of 07:09 on 8 July 2011. In Figure 8, the lines of ‘ + ’ are defined as ship sailing trajectories, ‘*’ is the position of the vessel at the initial moment, and ‘o’ is the end point of the vessel trajectory. Figure 8 shows the trajectories of multiple vessels in area A, which are approximately parallel or overlapping. The Maritime Mobile Service Identify (MMSI) number and tracking colours of the ships are as follows (here the symbol of ‘*’ is used to hide the full MMSI number for privacy and security reasons): MMSI number 26****280 designated by a green ‘ + ’, MMSI number 26****830 designated by a cyan ‘ + ’, and MMSI number 26****530 is designated by Red ‘ + ’. As shown in Figure 8(b), the green ‘ + ’ tracked a high-speed ship, and the red “ + ” was the navigation track of a cargo ship. Moreover, the majority of the ship trajectories in this area contain intersections.

Figure 8.

Multi-vessel sailing trajectories.

3.2. Possible relative striking location based on interval analysis

Based on the studies of ship encounter scenarios in the Baltic Sea in July 2011, possible relative striking locations can be determined using interval analysis. Figure 9(a) is the prediction of relative striking locations of two ships where the length ratio is approximately 0·83, and the MMSI numbers of the ships are 26****730 (red ‘ + ’) and 26****480 (blue ‘ + ’), respectively. The two ships sailed between 14:33 on 17 July 2011 and 14:55 on 17 July 2011. The relative angle, relative speed and speed of the two ships are shown in Figure 9(b). Ship 26****480 was faster than ship 26****730, and ship 26****480 arrived earlier than ship 26****730 at the intersection. The angle of navigation between the two ships was 79·9^°. Following this trend, a collision may occur between the two ships. According to Section 2.1.5, the possible relative striking location of ship 26****480 meets the requirements for a frontal collision and ship 26****730 is predicted to possibly collide with the rear-end of the ship.

Figure 9.

The prediction of possible relative striking location of ships 26****730 and 26****480 in the Baltic Sea at 14:35 on 17 July 2011. (a) The prediction of possible relative striking locations of two ships in the Baltic Sea, (b) Relative angle, relative speed and speed of two ships in the Baltic Sea.

3.3. Regional risk network

Section 2 summarises NVCRO after the qualitative analysis of vessel collision risks, which is an important part of the risk network analysis. Figure 10 presents a case where the value of NVCRO for four ships within 6 nm in the Baltic Sea, and the value of 10,478 in abscissa corresponds to the time of 06:37. Figure 10 depicts the trajectories of the vessels with MMSI numbers 23****840, 26****000, 27****000, 23****690 and the value of NVCRO in the Baltic Sea at 06:37 on 8 July 2011. As illustrated in the figure, the sailing trend of ship (magenta ‘ + ’), ship (cyan-blue ‘ + ’) and ship (blue ‘ + ’) were close to each other. The ship (red ‘ + ’) diverged from the other ships, thus its (23****690) NVCRO values are zero at 06:37.

Figure 10.

Vessel sailing trajectory map and VCRO of our vessels within 6 nm near Helsinki in the Baltic Sea.

Figure 11 shows the regional risk network within 6 nm, where ship (23****690) (corresponding to the cyan trajectory of Figure 10) diverged from ships 26****530, 26****730 and 26****480. The data values of the risk matrix of ship (23****690) were all zeros, which means it was navigationally safe in this area. Although ship 23****840 and ship 27****000 had a risk of near-miss collision with ship 26****000, the regional risk value was not high.

Figure 11.

Regional risk network near Helsinki in the Baltic Sea.

3.4. The assessment of regional collision risk

Marine collision risk of the chosen area in the Baltic Sea can be determined using the MVCRO and the density complexity (Wen et al., Reference Wen, Huang, Zhou, Yang, Xiao and Wu2015). The value of RVCRO represented actual situations of regional vessel collision risk. According to Equations (13)–(15), we analysed the total value of MVCRO and density complexity, which are shown in Figure 12. The value of MVCRO and density complexity were close to 19·7 and 27·9, respectively, which means a low collision risk based on the clustering results. The work presented here has been supported by AIS data, which validates the effectiveness of Figure 1 in Section 2 with a goal of assessing regional near-miss collision risk.

Figure 12.

Analysis of RVCRO, MVCRO and density complexity in an encountered scenario in the Baltic Sea.