2. Data

The present paper uses a high-quality traffic data-source consisting of AIS transmissions from 13 days covering the Norwegian exclusive economic zone (EEZ). AIS data dimensionality varies, but most sources provide traffic data for speed over ground (SOG), course over ground (COG), position (latitude, longitude), as well as unique vessel identifiers. The present data source additionally provides information on vessel type and dimensions. A typical AIS data set of the present quality and geographical coverage contains 18$\cdot$5 million registered AIS transmissions per day, with a mean temporal resolution of 9$\cdot$6 s. Rates of transmission vary with vessel speed and degree of course change, allowing for particularly high resolution when vessels are in manoeuvre. This temporal resolution further allows for detailed analysis of the aspects describing manoeuvring patterns, whereas the spatial coverage allows for analysing the manoeuvres over various traffic zones.

The data source has low rates of erroneous or missing records, at below 0$\cdot$1% for SOG and COG. Approximately 6$\cdot$0% of records are erroneously registered outside the area of analysis. To focus analysis on commercial vessels, records for which vessel category was not identified, or where vessels were identified as non-commercial, were not considered. In addition, tugs and pilot vessels were not considered, as such vessels by nature engage in close-quarters situations which are difficult to discern from NCSs. The ensuing typical filtered data set consists of 10 million daily records, which entails a sizeable number of vessel-to-vessel interactions to be analysed at each timestamp.

Using AIS data for analysis with uniquely identified vessels further allows for combination with other data sets, such as weather or vessel characteristics. The present paper focuses on suggesting a framework and highlighting some selected manoeuvre aspects, but the versatility of AIS data lends itself to numerous extensions.

3. Methodology

The following discusses the identification of real NCSs for an NCS database based on historical traffic data from AIS. The CPA framework is established by Sang et al. (Reference Sang, Yan, Wall, Wang and Mao2016) as a method for analysing the collision behaviour of two objects in motion. CPA is defined as the closest point two objects will arrive at if speed and course are unaltered. Distance between vessels at CPA (DCPA) indicates the severity of the hypothetical situation. Time to CPA (TCPA) is the remaining time for the two objects to reach CPA at constant speed and course. Negative TCPA indicates objects moving away from each other. An NCS is defined as a situation where two vessels will, in the near future, come within an unsafe distance of each other. NCSs are identified for each candidate situation by an unsafe DCPA – below a set threshold – and a limited positive TCPA.

AIS data provide positions, speed and course for a vessel $i$ at timestamp $t_0$, and hence provides the information

\[ t_0; \quad (x_i(t_0),y_i(t_0)) = (\textrm{LON}(t_0),\textrm{LAT}(t_0)); \quad \textrm{SOG}_i(t_0); \quad \textrm{COG}_i(t_0) \]

where $(x_i(t_0),y_i(t_0))$ is given in decimal degrees, $\textrm {SOG}_i(t_0)$ in knots and $\textrm {COG}_i(t_0)$ in degrees. The velocity vector is given by

(1)\begin{equation} V_i = \left[\begin{matrix} V_{i,x} \\ V_{i,y} \end{matrix} \right] = \left[\begin{matrix} \textrm{SOG}_i \sin(\textrm{COG}_i) \\ \textrm{SOG}_i \cos(\textrm{COG}_i) \end{matrix} \right] \times \frac{1{,}852}{3{,}600} \end{equation}

where all variables are evaluated at $t_0$. The $\textrm {TCPA}_{i,j}(t_0)$ can be determined for a pair of close vessels $(i,j)$ at time $t_0$ based on their relative velocity $\Delta V_{i,j}(t_0)$ and recorded relative distance $\Delta P_{i,j}^{0}(t_0)$. The $\textrm {TCPA}_{i,j}(t_0)$ is determined as $t-t_0$ for a $t$ such that the future distance $\|\Delta P_{i,j}(t)\|$ is minimised, for $t \geq t_0$. The relative distances are calculated as

(2)\begin{equation} \Delta P_{i,j}^{0} = \left[\begin{matrix} (x_i - x_j) \gamma \cos\left(\dfrac{y_i + y_j}{2}\right)\\ (y_i - y_j) \gamma \end{matrix} \right] \end{equation}

where $\gamma = 111{,}319{\cdot }9$ is the decimal degree-to-metre conversion rate, longitudes are curvature corrected and all expressions are evaluated at $t_0$. Relative velocity is calculated as

(3)\begin{equation} \Delta V_{i,j} = V_i - V_j \end{equation}

This gives $\textrm {TCPA}_{i,j}$ as

(4)\begin{equation} \textrm{TCPA}_{i,j} = \frac{-(\Delta P_{i,j}^{0})^{\top}\Delta V_{i,j}}{(\Delta V_{i,j})^{\top} (\Delta V_{i,j})} \quad \text{if} \ \Delta V_{i,j} \neq \left [\begin{array}{c} 0\\ 0 \end{array} \right] \end{equation}

If $0 \leq \textrm {TCPA}_{i,j}(t_0) \leq \textrm {TMAX}$, where the present paper lets $\textrm {TMAX} = 1{,}200$ s, we say vessels $(i,j)$ will be closest at $t = t_0 + \textrm {TCPA}_{i,j}(t_0)$ given present speed and course, and have

(5)\begin{equation} \Delta P^{\textrm{TCPA}}_{i,j} = \Delta P_{i,j}^{0} + \textrm{TCPA}_{i,j}\Delta V_{i,j} \end{equation}

and, further, the DCPA as

(6)\begin{equation} \textrm{DCPA}_{i,j} = \sqrt{(\Delta P^{\textrm{TCPA}}_{i,j})^{\top} (\Delta P^{\textrm{TCPA}}_{i,j})} \end{equation}

Otherwise, we define $\textrm {DCPA}_{i,j} = \infty$. A version of a function to determine $\textrm {TCPA}_{i,j}$ for two vessels may be implemented as

Listing 1. A simple function to determine TCPA for two vessels.

For $N$ vessels at time $t_0$, represented as explained in Section 2, the algorithm applies the function above to every pair of vessels using vectors instead of scalars. The function looks the same, and may be implemented asFootnote ²

Listing 2. Vectorised function to determine TCPA for N vessels.

where capital letters denote vectors.

The algorithm determines $\textrm {TCPA}_{i,j}$ and $\textrm {DCPA}_{i,j}$ for all vessel pairs $(i,j)$. An NCS is identified when $\textrm {DCPA}_{i,j}$ is found to be below a set (unsafe) threshold. The present paper identifies situations where the minimum DCPA is lower than three times the sum of the vessel lengths, i.e.

\[ \text{DCPA}_{\textrm{min}}(i,j) \leq 3 \times (L_i + L_j) \]

where $L_i$, $L_j$ are the vessel lengths.Footnote ³ The algorithm identifies this on the record level for each vessel pair. A database can be constructed in several ways. The present paper lets each NCS between a pair of vessels be identified by the record denoting the lowest $\textrm {DCPA}_{i,j}(t_0)$ for the vessels.

For every identified situation, full-resolution traffic data are retrieved from the original data set for both vessels at all timestamps preceding and succeeding the timestamp of minimal $\textrm {DCPA}_{i,j}$ by a sufficient time span, to further analyse the CPA and manoeuvre patterns during the NCS. Choice of time span must be sufficiently long to encompass the situation, and must be informed by the size and type of the vessels. The present paper retrieved records 60 min prior to and following the point of minimal $\textrm {DCPA}_{i,j}$.

Transmission times differ, and hence traffic information must be synchronised for each pair of vessels participating in a situation to analyse CAMs. Records are synchronised by a process where each record, with timestamp $t_{0,i}$, is coupled with an equivalent $t_{0,j}$ in the other vessel's timestamp vector such that $|t_{0,i,k} - t_{0,j,l}| < 5$ and $t_{0,j} = \arg \min _{t_{0,j} \in T_0}|t_{0,i} - t_{0,j,k}|$, where $T_0$ is the vector of timestamps for vessel $j$. Records which have no unique equivalent are discarded. The traffic information synchronisation is implemented as

Listing 4. Function to synchronise observations

where the records are organised such that T0j is the longest timestamp vector of the two.

The synchronised time for each record is taken to be the mean timestamp of the coupled records. This introduces imprecision, but the common time scale is not used for computations. At this stage, both vessels are observed at approximately identical timestamps. The synchronised traffic data for each situation are then used to analyse the traffic patterns during CAMs. First, the timestamp when the CAM is initiated is detected as $t_{1,i,j}$ and the timestamp when the NCS is resolved is detected as $t_{F,i,j}$. The evacuation time window is defined as $[t_{1,i,j},t_{F,i,j}]$ for each situation, according to the following rules.

1. 1. If $\textrm {DCPA}_{i,j}$ does not pass below 10 m, the time window is initiated at the minimum $\textrm {DCPA}_{i,j}$.Footnote ⁴ This is the normal situation where a CAM is initiated to ensure a safe passing distance by increasing the (predicted) DCPA.

2. 2. If $\textrm {DCPA}_{i,j}$ passes below 10 m, the time window starts at the initial point where $\textrm {DCPA}_{i,j}$ passes below 10 m for which $\textrm {TCPA}_{i,j}$ has not turned negative. For such NCSs, the calculated values are assumed to be subject to noise, and the first crossing of 10 m is considered the global minimum.

3. 3. If $\textrm {DCPA}_{i,j}$ passes below 10 m, but $\textrm {TCPA}_{i,j}$ is negative at this point, the time window starts when $\textrm {DCPA}_{i,j}$ is at its minimum while $\textrm {TCPA}_{i,j}$ has not turned negative.

4. 4. If $\textrm {DCPA}_{i,j}$ never passes below a threshold $\textrm {DMAX}$ metres (in the present paper set as three times the sum of the vessel lengths), as explained previously, the situation is discarded from the NCS database.

If the situation is confirmed as an NCS, it is taken to be resolved at $t_{F,i,j}$, the first point after $t_1$ when $\textrm {TCPA}_{i,j}$ turns negative. This implies that the two vessels involved in the NCS are no longer approaching one another. The time interval $t_{F,i,j} - t_{1,i,j}$ is an important manoeuvre aspect as it determines how ‘early’ (as emphasised in COLREGs) CAMs were initiated, and how the linguistic variable ‘early’ from COLREGs is interpreted in real situations.

An observed CAM is defined for a vessel if, in a situation, the vessel alters her course or speed by a significant (observable) amount during the evacuation time window. For each situation, we observe either no vessel, one vessel or both vessels engaged in a CAM. In an ideal application, this determines which vessel should take responsibility to give-way and which vessel practised the right-of-way and remained in a stand-on state, as per COLREGs. Furthermore, for each vessel, we may observe no CAM, a course alteration, a speed alteration or both a speed and course alteration. If one of the latter three scenarios is recorded, the vessel is defined as having implemented a CAM. Each of the latter three scenarios defines a type of CAM. Situations where one or more vessels engage in a CAM are the NCSs of interest. Some speed change is also observed during course alteration manoeuvres, as shown in Figure 1. In practice, large vessels more often engage in course-change manoeuvres rather than incremental speed-change manoeuvres. Combining this assumption with Figure 1, one may deduce that a course-change CAM will often entail a detectable speed change and, hence, be registered as a combined manoeuvre. This understanding should enter into the researcher's analysis of the results.

Figure 1. Joint distribution of the magnitude of observed steady-state course and speed change for vessels where a course-change CAM is observed. The figure indicates that there is a speed change observed for course-change CAMs

Classification of manoeuvre type is done vessel-by-vessel, by analysing the individual patterns of changes in COG and SOG over the entire CAM time window, as shown in Figure 2. A course-change CAM is classified as such if the time derivative of the COG deviates significantly from the measurement noise inherent in the time series. The same method is used for classifying speed-change CAMs, by exchanging COG with SOG. In this regard, the threshold for ‘measurement noise’ is defined as 25% above the maximum absolute value of the COG time derivative for the 60 s prior to the start of the evacuation window. If the COG time derivative passes above this measurement noise threshold, the vessel is classified as having engaged in a course-change CAM. The application is illustrated for a representative vessel in Figure 2(a). The figure shows how a threshold is defined based on the pre-$t_1$-period, and that a course-change CAM is defined for this vessel, as the within-window COG time derivative passes above this threshold. In addition to being useful for analysing the practice of the COLREGs rules, these aspects allow for the analysis of which and how many types of manoeuvres are executed to avoid collision in each situation.

Figure 2. COG time derivative for two representative vessels in an overtaking situation: (a) successful classification; (b) faulty initial classification, needs correction. The black line shows the maximum noise recorded prior to the evacuation window. The red line shows the implied noise threshold. For both vessels, a course-change manoeuvre is initially registered. In the first example, this is also the final classification. The grey-shaded field indicates the time window within which the noise threshold is defined. In the second example, our methodology recognises that the total registered course change is too small to indicate an actual course-change manoeuvre, and the final classification is corrected to ‘No manoeuvre’. The time series is taken from the NCS example presented in Figure 7

This method is not without its limitations. Measurement noise is amplified when calculating derivatives, and the choice of 25% and 60 s as tuning parameters should be subject to refinement. As seen in Figure 2(b), if the time derivative is consistently and sufficiently close to zero for the duration of the situation, the measurement noise threshold may be set too low, and a CAM may incorrectly be recorded. In particular, misclassifications should be expected for vessels where the total speed change is close to zero. To correct for such misclassifications, vessels for which no observable steady-state course change is recorded are classified as not having performed a course-change CAM.

The steady-state course change for a vessel is defined as the difference between the mean COG for 11 observations surrounding the start of ($t_1$) and end of ($t_F$) the evacuation time window.Footnote ⁵ We define the observed course change as being observationally zero if it is below $1{\cdot }5^{\circ }$. This roughly conforms to the lower $30$th percentile of the distribution of observed steady-state course change for all vessels. The same approach is undertaken with respect to speed change, but where the threshold for what is considered observationally zero is set at below 0$\cdot$15 knots of change, which conforms approximately to the lower $30$th percentile of the distribution.Footnote ⁶ These corrections result in a change in the course-change CAM status for 6$\cdot$2% of all NCS-involved vessels, and for 9$\cdot$6% of all vessels with regards to the detection of a speed-change CAM. The distribution of situation types, differentiated by manoeuvre status, is presented in Figure 3.

Figure 3. Distribution of registered manoeuvre types for NCSs where at least one CAM is initiated. In the case where both types of manoeuvres are registered, the intuition is not that both manoeuvres were necessarily intended, but rather that a course change often entails a speed change

The observed steady-state change in COG and SOG are also important indicators of how ‘substantial’ the course and/or speed alterations are in real situations. This allows for evaluating the real interpretation of ‘substantial’ in terms of COLREGs compliance for each situation.

The full comprehension of CAMs further requires analysis of the conditions at which the manoeuvres are initiated and their effect on solving the NCS. Several manoeuvre aspects may be of interest. The present paper estimates the relative approach speed, the distance between vessels at $t_1$ and passing distance, in addition to the evacuation time, and the steady-state course and speed change for different situations in aggregate. The aspects are defined as follows.

• • Relative approach speed is computed as the absolute value of the relative velocity, $|V_i - V_j|$, between the two vessels in the first half of the evacuation time window, $[t_1,t_M]$ (where $t_M = (t_1 + t_2) /2$). This indicates the speed with which the vessels are approaching each other.

• • Vessel distance at $t_1$, $|P(t_1) - Q(t_1)|$, is computed as the actual distance between the two vessels at the start of the CAM, $t_1$, and it measures how close vessels are before initiating a CAM.

• • Passing distance, $|P(t_F) - Q(t_F)|$, is computed as the actual distance between vessels at the end of the evacuation time window, $t_F$. This measures the effect of the CAM in avoiding close-quarters situations.

• • Evacuation time, $|t_F - t_1|$, is calculated as the duration from the start of the CAM to the end of the NCS. This indicates how early the manoeuvre was initiated.

• • Steady-state course (respectively speed) change is calculated for each vessel as the difference between the mean COG (respectively SOG) for 11 observations around the start and end of the evacuation time window.

To analyse situations with respect to practised CAMs, all NCSs are classified based on which COLREGs rule applies for the vessels. In particular, COLREGs rules 13, 14 and 15 stipulate actions to be taken by the participating vessels in situations where a vessel is overtaking, crossing or in a head-on encounter (Ventura, Reference Ventura2005).

COLREGs rule 13(b) specifies that ‘A vessel shall be deemed to be overtaking when coming up with another vessel from a direction more than $22{\cdot }5^{\circ }$ abaft her beam (…)’, whereas COLREGs rule 14(a) specifies that a head-on situation takes place ‘When two power-driven vessels are meeting on reciprocal or nearly reciprocal courses so as to involve risk of collision (…)’ and 14(b) that ‘Such a situation shall be deemed to exist when a vessel sees the other ahead or nearly ahead (…)’. Conditions for when vessels are in a crossing situation (with actions as required by rule 15) are not specified, but it is generally understood to be situations which do not fulfil the criteria of rules 13 and 14.

To classify NCSs in light of COLREGs, it is necessary to mathematically define the terms used in the different rules. As any subsequent alteration of course or bearing should not change the class of a situation, a situation is classified based on its traffic features prior to $t_1$, i.e. before the vessels initiate any CAMs. Data on vessel heading are often recorded as part of AIS, but are in the present data considered to involve too many erroneous and missing records to be used as part of a robust measure. Hence, the heading of a vessel $i$ at $t_1$ is approximated as the mean course in the observations preceding $t_1$,

(7)\begin{equation} \overline{\textrm{COG}_i} = \operatorname{mean}(\textrm{COG}_i(t))) \quad \text{s.t.} \ t_1-20 \leq t \leq t_1 \end{equation}

Taking the mean over several records makes the measure more robust against noise and contributes to the accuracy of the classification.

In any encounter between two vessels $i$ and $j$, the relative angle of approach, henceforth denoted $\Delta \textrm {COG}$, is determined as the difference, translated to the interval $[0^{\circ },360^{\circ })$,

(8)\begin{equation} \Delta \textrm{COG} = \textrm{mod} (\overline{\textrm{COG}_i} - \overline{\textrm{COG}_j},360^{{\circ}}) \end{equation}

The relative bearing between vessels $i$ and $j$ with respect to $i$ is calculated as

(9)\begin{equation} \beta_i = \textrm{mod}(\text{angle}(\Delta P_{j,i}^{t_1}) - \overline{\textrm{COG}_i},360^{{\circ}}) \end{equation}

where $\text {angle}(\Delta P_{j,i}^{t_1})$ is the angle of the relative distance vector starting from the position of vessel $i$ and ending at the position of vessel $j$ at $t_1$, as defined in Equation (2). This angle is the absolute bearing and is measured with respect to the north in a clockwise direction.

Correspondingly, $\beta _j$ is the relative bearing with respect to vessel $j$ and is calculated by switching the indices $i$ and $j$ in Equation (9). If vessel $i$ is own ship and vessel $j$ is target ship, $\beta _i$ and $\beta _j$ are generally understood to be the bearing angle and contact angle, respectively, see Woerner et al. (Reference Woerner, Benjamin, Novitzky and Leonard2019).

NCSs are classified based on $\beta _i$, $\beta _j$ and $\Delta \textrm {COG}$, calculated at $t_1$. Starting with rule 13(b), a vessel $j$ is overtaking vessel $i$ if $j$ has a relative bearing of more than $\phi _o$ behind the beam of vessel $i$, equivalently $90^{\circ } + \phi _o$ off her centreline, where $\phi _o =22{\cdot }5^{\circ }$ is clearly defined in COLREGs.

Rule 14(a) specifies that two vessels are head-on if $\Delta \textrm {COG}$ is in the interval $180^{\circ }\pm \phi _h$. Here $\Delta \textrm {COG} = 180^{\circ }$ indicate reciprocal courses, and $\phi _h$ is an added tolerance for ‘nearly’ reciprocal. The $\phi _h$ further accounts for the qualification in rule 14(c) that ‘When a vessel is in any doubt as to whether such a situation exists, she shall assume that it does exist and act accordingly’.

A head-on situation might equivalently be classified using $\beta _i, \beta _j$ and rule 14(b). In this case, both relative bearings should be close to $0^{\circ }$, such that the vessels see each other ‘ahead or nearly ahead’. The former approach is chosen in the following analysis.

Neither part of rule 14 specifies numerically the magnitude of $\phi _h$. Court rulings indicate a convergence towards understanding ‘nearly reciprocal’ as meaning $\phi _h \in [5,6]$. Recent additions to the literature are however not unanimous as to how $\phi _h$ should be interpreted. Woerner et al. (Reference Woerner, Benjamin, Novitzky and Leonard2019) use $\phi _h=13^{\circ }$ as their default tolerance angle for ‘reciprocal or nearly reciprocal courses’, whereas Wang et al. (Reference Wang, Yu, Liang and Li2018) apply $\phi _h=15^{\circ }$ when studying obstacle avoidance. At the upper end, Li et al. (Reference Li, Zhang, Liu and Zhang2020) and Cho et al. (Reference Cho, Han and Kim2020) apply $\phi _h=22{\cdot }5^{\circ }$ in their analysis. It is outside of the scope of this paper to conclude as to the true value of $\phi _h$. However, the noisy properties of AIS data argue against opting for the narrowest definition. In the following, $\phi _h = 10^{\circ }$ is applied in the classification algorithm.Footnote ⁷

A situation is mathematically defined by the following conditions: If a situation is an NCS (where collision risk exists), we have the rule applying as

\[ \cdots=\begin{cases} j \text{ overtaking } i & \text{if } \cos(\beta_i) \leq \cos(90+\phi_o) \\ i \text{ overtaking } j & \text{if } \cos(\beta_j) \leq \cos(90+\phi_o) \\ \text{Head-on} & \text{if } \cos(\Delta \textrm{COG}) \leq \cos(180+\phi_h) \\ \text{Crossing} & \text{Otherwise} \end{cases} \]

where cosines are used as they treat angles as equal on each side of $180^{\circ }$ and $0^{\circ }$.