International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 5

Number 4

December 2011

497

1 LITERATURE REVIEW

A number of models for ship-ship collision probabil-

ity estimation can be found in the literature. They

can be divided into two major groups namely: static

and dynamic models. The static models are simpler

and less time consuming for computation, however

their accuracy can be questioned. The dynamic

models are more complex, in principle they need

more input data than static models, but their results

are comprehensive. In this section the short over-

view of existing models will be made, and our con-

tribution to the existing knowledge will be put for-

ward.

1.1 Static models

The most known approaches were introduced by

(Fujii et al. 1970) and (MacDuff 1974). Models of

this kind have been commonly used in recent dec-

ades and won the popularity among researches main-

ly due to simplicity and robustness. However they

have also some drawbacks, lack of ship dynamics or

assumption regarding a collision between two ships.

A collision is defined as a meeting of two ships in a

distance named the ”collision diameter”, which

means almost the physical contact. Such an assump-

tion may lead to an understanding that in any ship-

ship encounter at a distance greater than the ”colli-

sion diameter” these ships are able to avoid a colli-

sion, which in most cases is not true. Despite the

drawbacks the model was adopted by (Pedersen

1995), and with minor modifications was used to de-

termine the safety of navigation in many European

waters: (Otto et al. 2002), (Sfartsstyrelsen 2008).

Hence in Europe it is mostly known as Pedersen

model. Another method for the frequency of colli-

sion estimation, making an assumption regarding

uncorrelated traffic, was outlined by (Fowler and

Sorgrad 2000). A critical situation is assumed to oc-

cur when ships come to close quarters to a distance

of 0.5 Nm of each other, which is constant regard-

less of a meeting scenario. A model for encounter

probability estimation proposed by (Kaneko 2002)

defines a critical area of an optional form of a closed

boundary, around a ship which violation means col-

lision. Kaneko in his model recognizes two shapes

of the critical area: rectangular and circular, but

again the size of the area is fixed. A series of papers

utilizing the ship domain approach to ship-fixed ob-

ject collision assessment was published also by

(Gluver and Olsen 1998) and (Pedersen 2002).

However, none of the model listed above takes

ship dynamics into consideration.

1.2 Dynamic models

Another group of models utilize marine traffic simu-

lations. A group of researches led by Merrick pro-

posed a risk analysis methodology for maritime traf-

fic in coastal areas based on system simulation

A New Definition of a Collision Zone For a

Geometrical Model For Ship-Ship Collision

Probability Estimation

J. Montewka ‡, F. Goerlandt & P. Kujala

‡Aalto University, School of Engineering, Finland; Maritime University of Szczecin, Poland

Aalto University, School of Engineering, Finland

ABSTRACT: In this paper, a study on a newly developed geometrical model for ship-ship collisions probabil-

ity estimation is conducted. Most of the models that are used for ship-ship collision consider a collision be-

tween two ships a physical contact between them. The model discussed in this paper defines the collision cri-

terion in a novel way. A critical distance between two meeting ships at which such meeting situation can be

considered a collision is calculated with the use of a ship motion model. This critical distance is named the

minimum distance to collision (MDTC). Numerous factors affect the MDTC value: a ship type, an angle of

intersection of ships’ courses, a relative bearing between encountering ships and a maneuvering pattern. They

are discussed in the paper.

498

(Merrick et al. 2002), (Merrick et al. 2003). Mari-

time traffic is simulated in the time domain based on

routes obtained from expert opinion and vessel arri-

val records. Finally, these were combined with the

simulation output in order to carry out a risk analysis

(van Dorp and Merrick 2009).

Another probabilistic model for the assessment of

navigational accidents in an open sea area was out-

lined by (Gucma and Przywarty 2007). The method

makes use of a simplified model of maritime traffic,

which is simulated in the time domain. A recent

model, introduced by (Goerlandt and Kujala 2011) is

based on an extensive time-domain simulation of

maritime traffic in a given area. Vessel movements

are modelled based on data obtained from a detailed

study of route-dependent vessel statistics. The colli-

sion candidates are detected by a collision detection

algorithm which assesses the spatio-temporal propa-

gation of the simulated vessels in the studied area.

Markov, semi-Markov and Random Field theory

based models for maritime traffic safety estimation

were introduced recently (Smalko and Smolarek

2009), (Smolarek and Guze 2009), (Smolarek 2010),

(Guze and Smolarek 2010). However the main as-

sumption of the models proposed is that traffic flow

is stationary, which is not applicable to areas with

scheduled traffic. Recently a geometrical model for

estimation the probability of ship collisions while

overtaking were introduced by (Lizakowski 2010),

his model considers human factor and the fairway

and ship dimensions. However all these models are

advanced mathematically they do not take into ac-

count ship dynamics nor human factors.

A multicomponent model for an inland ship safe-

ty estimation was presented by (Galor 2010). How-

ever each of the proposed model’s component is es-

sential, the model itself constitutes rather an

introduction to the further quantitative analysis of

the problem.

For the first time the idea of ship manoeuvrability

implementation into a collision assessment model

was presented by (Curtis 1986). However, this mod-

el was limited to one ship type, which was a very

large crude carrier (VLCC), and only overtaking and

head-on situations were considered.

1.3 Authors’ contribution

A new criterion for ship-ship collision probability

estimation and a new model have been introduced

by (Montewka et al. 2010). The model considers

ship maneuverability and traffic parameters; the new

collision criterion is named the Minimum Distance

To Collision (MDTC). MDTC is a critical distance

between two ships being on collision courses, at

which they must perform collision evasive actions,

in order to pass safely. The MDTC is estimated by

means of ship motion model and series of experi-

ment for various ship meeting scenarios.

This paper is a continuation of our previous re-

search, it consists of the detailed analysis of the

MDTC values for a wide ranges of input variables

(they are defined in the following Chapters) and two

patterns of performing collision evasive action. The

maneuvering patter one means that own ship is per-

forming a collision evasive action and the other ship

is not acting, in the maneuvering patter two both

ships are involved in avoiding a collision. Perfor-

mance of turning circle is considered a collision eva-

sive action.

2 INTRODUCTION TO MDTC MODEL

The MDTC model introduced in a previous work of

(Montewka et al. 2010) and developed further in this

paper, is based on an initial assumption, that two

ships collide if the distance between them becomes

less than a certain value, named a MDTC. This

MDTC value is not a fixed number, but it is calcu-

alted dynamically for each type of vessel and en-

counter individually. Thus it changes with the situa-

tion. The main factors affecting the MDTC value

are: the vessels maneuverability, the angle of inter-

section labelled α in Figure1a, the relative bearing

from one vessel to the other labelled β in Figure1b

and a pattern of evasive maneuvers (one vessel

swinging or both). In the previous study, a simpli-

fied methodology was applied, which assumed that

two vessels met at a constant relative bearing while

proceeding with their service speeds. Presented

study considers a wide range of relative bearings,

varying from 10 do 80 degrees (counting from the

own ship’s bow) and takes into account two differ-

ent engine settings for each ship type, therefore

providing more detailed results.

Figure 1: A definition of MDTC and major factors affecting it

Source: (Montewka et al. 2010)

3 RESEARCH MODEL

The theory of the model and preliminary research

aiming to define the ”collision zone” were presented

by (Montewka et al. 2010). In this paper the results

of studies with respect to different ship types and

499

ship speeds and varying meeting angles are shown.

However only planar motion of a ship is taken into

account and assumption regarding ship navigating

through deep water is made. We also assume, that

the prevailing weather conditions do not deteriorate

significantly the maneuverability of ships sailing in

the analyzed part of the Gulf of Finland. In order to

validate it, we simulated a maneuver of turning cir-

cle to starboard side, for the chosen ship type, which

was a RoPax (for ship particulars see Table 1), for

two different wave conditions (no wave, and an av-

erage wave height for the Gulf of Finland). Accord-

ing to (Pettersson et al. 2010) and (Raamet et al.

2010) the average monthly weave height recorded in

the analyzed area (sea between Helsinki and Tallin)

does not exceed 2 meters, and as a such was adopted

for the simulation. For this purpose the Laidyn ship

motion model was adopted (Matusiak 2007).

The results allowed us to keep our assumptions,

as a difference between the trajectories of a ship in

two different heights of a wave seems to be negligi-

ble for the purposes of our research (Figure 2).

Figure 2: Turning circles of RoPax performed for two different

wave heights

3.1 Ships considered

In the course of our analysis we are considering four

major ship types: a passenger ship, a containers car-

rier, a RoPax and a tanker. In each scenario, ships

are assumed to proceed with two different engine

settings (except for a passenger vessel which is as-

sumed to sail always at a maximum speed) which re-

sult in forty two encountering scenarios, as depicted

in Figure 4. The following abbreviations are used:

’FA’ is full ahead and ’HA’ means half ahead. The

’FA’ abbreviation corresponds to a mean speed of a

ship of given type as obtained from recorded AIS

data. The abbreviation ’HA’ does not correspond to

an actual engine setting, it rather reflects a spread of

recorded speed values for a given class of ships in

the analyzed area. The value of ’HA’ for given ship

type was calculated by subtracting the standard de-

viation from the mean value for a given type of ship.

The main particulars of the analyzed vessels are

listed in Table 1.

Table 1: Ships particulars.

___________________________________________________

Ship type LOA B T v

[m] [m] [m] [kn]

___________________________________________________

Container carrier 150.0 27.2 8.5 20;17

RoPax 158.6 25.0 6.1 20;18

Tanker 139.0 21.0 9.0 14;11

Passenger 185.0 27.7 6.5 25

___________________________________________________

3.2 Encountering scenarios

Each of an encountering scenario is run for seven-

teen different crossing angles (α), varying from 010

to 170 degrees with 10 degrees increment. Where

010 degrees means almost overtaking (vessel B on a

course of 350deg) and 170 stands for almost head-on

meeting (Vessel B on a course of 190deg), as de-

picted in Figure 3. The situation shown there consid-

ers own ship seeing another at 45 degrees relative

bearing. In the course of the experiment, each cross-

ing angle is calculated for a range of relative bear-

ings, from 10 to 80 degrees, counting from the own

ship’s bow.

Figure 3: Relative positions of vessels, with three chosen cross-

ing angles, before they start to maneuver, (Montewka et

al.2010)

For each ship-ship encounter at a given intersec-

tion angle (α) and at a given relative bearing (β), one

MDTC value is obtained. As specified in a block di-

agram depicted in Figure 4, in total 5712 MDTC

values are obtained. Then for each intersection angle

(α) the maximum MDTC value among eight (as

there are eight relative bearings considered) is

drawn. Also the relative bearing which is the most

inconvenient from a collision evasive point of view,

and which requires the most space to make an action

is indicated. For further statistical analysis 714 out

of all 5712 MDTC values are selected for each ma-

neuvering pattern.

500

3.3 Maneuvering patterns

In case of a maneuvering pattern number one, the

own ship is performing an evasive action, by turning

circle, and another ship is following her initial

course. In case of maneuvering pattern number two,

two ships are performing turning circles in order to

avoid collisions. The following simplifications in the

presented methodology are done:

− in case of evasive pattern where two vessels per-

form turning circles, they both start their maneu-

vers at the same time;

− ships are turning away from each other, which

implies course alteration away from each other to

avoid collision and to shorten the time at close

quarters (such assumption meets requirements of

the COLREG, which states, that ships must avoid

altering courses towards each other if in close

quarters);

− the settings of ships’ engines and rudders are con-

stant during maneuvers;

− the influence of weather conditions is omitted;

− the hydromechanical ship-ship interactions are

omitted.

Figure 4: Research model

3.4 MDTC estimation

In order to calculate the value of MDTC for a given

pair of vessels, an iterative algorithm is used, as de-

picted in Figure 6. The basic assumption is that the

two ships collided at a time instant t0. Then starting

from this time the reverse iterative algorithm is ap-

plied. It uses backward calculation method in a

space-time domain. Two trajectories of two ships are

drawn and the consecutive positions of ship’s centre

of gravity are plotted every second (dt=1s). If corre-

sponding ships’ contours following the trajectories

have at least one common point, indicating that they

both collided, the algorithm increases the initial dis-

tance between these two ships by constant value of

0.1LOA

average

. The trajectories are redrawn starting

from the new initial positions of the ships. This pro-

cess is repeated until the two contours of ships have

no overlaps at any time instant for a given relative

bearing.

New initial positions are defined by moving ship

B from ship A away. For a given meeting scenario (a

given angle of intersection α and a given relative

bearing β) the ships are moved away along a line of

a given relative bearing (β line). For the simplicity

of calculations it is assumed, that own ship holds her

initial position, while the other ship is moved away

along the β line.

In the situation where two trajectories have no

common points and the contours of the ships do not

over-lap, the initial position of vessel B is recorded

(as the initial position of own ship A was always

(0,0)), and the distance between these two positions

is calculated and stored. This distance, is named

MDTC for a given relative bearing. As each meeting

scenario is analyzed for a range of relative bearings

(from 10 to 80 degrees), the procedure presented is

repeated for all relative bearings, yielding eight val-

ues of MDTC for each angle of intersection α. Final-

ly, the maximum value of MDTC among these eight

is drawn. This maximum value is considered a

MDTC value for a given angle of intersection. This

procedure is repeated for all angle of intersection,

then for each maneuvering patterns. Thus the MDTC

charts are obtained.

In order to determine the MDTC charts, all rou-

tines are encoded in MATLAB. As a polygonal re-

gion, which could represent a ship contour an ellipse

is chosen. To determine, whether two contours of

the ships (represented as ellipses) overlap, the fol-

lowing MATLAB function is applied (MathWorks

2010):

IN = inpolygon(X,Y,xv,yv), (1)

it returns a matrix IN of the same size as X and Y.

Each element of (IN) is assigned the value 1 or 0

depending on whether the point (X(p,q),Y(p,q)) is in-

side the polygonal region whose vertices are speci-

fied by the vectors xv and yv.

For the sake of computation effectiveness each el-

lipse is transformed into discrete form and the num-

ber of points that represent the ellipse is 24. The el-

lipse’s axes are defined in the following way:

a = 0.5LOA

b = 0.5B, (2)

where a denotes a major axis, b is a minor axis, LOA

means length overall of a ship and B is a ship’s

breadth.

A MDTC value for a given encounter implies a

safe passage of two vessels, which corresponds to a

situation where these two vessels approximated by

the ellipses, will always be separable and will not

touch each other at any time step of a collision eva-

501

sive action. A graphical interpretation of above is

depicted in Figure 5, where both ships are at the

closest distance in the time step 81sec., however

they are still separable. A block diagram showing an

algorithm applied in the study to estimate a MDTC

chart for a given meeting scenario is depicted in

Figure 6.

4 DATA ANALYSIS

The data obtained in the course of MDTC calcula-

tions (see Figure 6) considers different ship types,

different engine settings and two different maneu-

vering patterns, as stated in Figure 4. In the next step

the statistical analysis of the obtained data is per-

formed.

Figure 5: Ships as ellipses and interpretation of a non-contact

passage

In Figure 7 data sets concerning MDTC assuming

a maneuvering pattern number one (own ship in-

volved in collision evasive action), according to a

ship type, are presented. Whereas the data depicted

in Figure 8 shows appropriate values for MDTC, ac-

cording to a ship type for maneuvering pattern num-

ber two (both vessels are involved).

To determine whether the differences which can

be noted visually are significant from the statistical

point of view, the appropriate statistical tests are per-

formed. The following are hypothesized:

− H

: the obtained values of MDTC, for a range of

intersection angles α, are drawn from the same

population (or equivalently, from different popu-

lations with the same distribution), thus MDTC

do not depend on a ship type.

− H

: the medians of analyzed variables are not all

equal, thus the MDTC values do not originate

from the same population, and they are a ship

type dependent.

In order to validate these hypotheses we performe

a nonparametric Kruskal-Wallis test which compares

samples from two or more groups, as the obtained

data do not follow a normal distribution. In the case

presented here we analyze 42 different encounters,

each consisting of 17 crossing angles, as depicted in

Figure 4. We form a 42-by-17 matrix, where each

column of the matrix represent an independent sam-

ple containing 42 mutually independent observa-

tions, and a number of columns is equivalent to a

number of crossing angels α. The function that

Kruskal-Wallis test is based on compares the medi-

ans of the samples in a matrix, and returns the p-

value for the null hypothesis.

In the course of the analysis the obtained p-value

vary for two maneuvering patterns concerned. In the

case where both vessels make a turn (the maneuver-

ing pattern No 2) the p-value yields 0.9988. This

shall not cast any doubt on the null hypothesis, and

suggests that all sample medians come from the

same population.

Figure 6: Block diagram for MDTC calculation

However, the results obtained for the maneuver-

ing pattern No 1, where the own ship performs colli-

sion evasive action only, are more scattered there-

fore not so straightforward in inference. The results

of the statistical tests concerning both maneuvering

patterns are gathered in Table 2. Analyzing a full

range of intersection angles, for maneuvering pattern

No 1, there is no evidence for not rejecting the null

502

hypothesis. This can lead to thinking that at least one

data set originates from a different population that

the other data sets.

Table 2: Results of statistical tests ordered by the angle of in-

tersection

___________________________________________________

Maneuvering Segment α p Hypothesis

pattern [deg] value value rejected

___________________________________________________

No 1 10-170 0.05 0.002 H

No 2 10 − 170 0.05 0.9988 H

___________________________________________________

However dividing the intersection angles range

into segments, and analyzing them separately, makes

it feasible to defend the null hypothesis.

In order to made a cross check of the results ob-

tained in this analysis, in the next step we order a da-

ta set according to a ship type, and run the Kruskal-

Wallis tests on smaller samples. The results obtained

are presented in Table 3.

Table 3: Results of statistical tests according to a ship type - the

maneuvering pattern No 1.

___________________________________________________

Ship type Segment α p Hypothesis

[deg] value value rejected

___________________________________________________

Container 10-170 0.05 0.4424 H

RoPax 10-170 0.05 0.9891 H

Tanker 10-170 0.05 0.1237 H

Passenger 10-170 0.05 0.9307 H

___________________________________________________

Figure 7: MDTC values for a given ship type - maneuvering

pattern No 1 (own vessel performing an evasive action)

Figure 8: MDTC values for a given ship type - maneuvering

pattern No 2 (both vessels performing collision evasive ma-

neuvers)

It can be noticed that two cases (”RoPax” and

”Passenger”) defend the null hypothesis for a full

range of intersection angles. In a case of ”Container”

the null hypothesis can not be rejected, but the

p − value obtained is not as high as in the previous

cases, however still acceptable. In a case of ”Tank-

er”, although the p − value obtained is greater than

the adopted level α, however the null hypothesis can

be rejected as early as a confidence level α is 0.13.

This obviously can cast some doubts on the null hy-

pothesis. In order to have an insight into a data set

regarding variable ”Tanker” we divide a range of in-

tersection angles into three segments, and run again

the statistical test. The results obtained are presented

in Table 4.

Table 4: Results of statistical tests according to a ship type and

the angle of intersection - the maneuvering pattern No 1.

___________________________________________________

Ship type Segment α p Hypothesis

[deg] value value rejected

___________________________________________________

Tanker 10-120 0.05 0.4411

Tanker 120-140 0.05 0.6671

Tanker 140-170 0.05 0.8306

___________________________________________________

Having variable ”Tanker” excluded the p-value

for the null hypothesis for the maneuvering pattern

No 1 becomes higher than adopted level α.

Table 5: Results of statistical tests according to the angle of in-

tersection, the maneuvering pattern No 1, tankers excluded.

___________________________________________________

Maneuvering Segment α p Hypothesis

pattern [deg] value value rejected

___________________________________________________

No 1 10-170 0.05 0.20

No 1 10-120 0.05 0.98 H

No 1 120-140 0.05 0.70

No 1 140-170 0.05 0.39

___________________________________________________

The following conclusions can be made:

503

− for the maneuvering pattern No 1, if the variable

”Tanker” is excluded, the statistical tests prove,

that the data for other three variables (”Contain-

er”, ”RoPax” and ”Passenger”) are drawn from

the same population;

− for maneuvering pattern No 1, the MDTC values

for tankers are obtained in the course of a sepa-

rate analysis;

− for maneuvering pattern No 2, there are strong

evidences, that all data come from the same popu-

lation, thus MDTC is not a ship type depended

variable.

5 RESULTS

The obtained MDTC values are categorized accord-

ing to an intersection angle α, and we make attempts

to define distributions of MDTC values for each an-

gle α and for given maneuvering patterns. Because

of the limited survey sample, and data scatter none

of commonly known distributions, neither continu-

ous nor discreet, fit the data. Thus further analysis is

conducted using one of the sampling methodology,

namely a non parametric bootstrap procedure.

For each maneuvering pattern, the following val-

ues are estimated: a mean and a standard deviation

of a MDTC for a given angle of intersection (α). In

order to obtain these parameters, the following pro-

cedure is adopted (Vose 2008):

− to collect the data set of n samples {x

…x

} in our

case n=42

− to create B bootstrap samples {x

*…x

*} where

each x

is a random sample with replacement

from {x

...x

}, in our case B = 10

;

− to estimate, for each bootstrap sample {x

...x

the required statistics sˆ. The distribution of these

B estimates of s represents the bootstrap estimate

of uncertainty about the true value of s.

The outcome of the bootstrap analysis are as fol-

lows:

− the mean MDTC values, for each intersection an-

gle α;

− the 0.95 confidence intervals around the mean

values (represented as dotted lines in a Figure 6).

Then using the upper confidence interval of the

mean value, the 0.95 prediction interval is calculat-

ed. The upper prediction band obtained is shown as

a solid line with diamonds. The results obtained, for

two maneuvering patterns are depicted in the follow-

ing Figures: 9, 10, 11.

Figure 9: The obtained MDTC chart for the maneuvering pat-

tern No 1

Figure 10: The obtained MDTC chart for tankers - the maneu-

vering pattern No 1

Figure 11: The obtained MDTC chart for the maneuvering pat-

tern No 2

6 CONCLUSIONS

This paper addresses a chosen aspects of marine

traffic safety modelling. This means a novel method

for definition a collision-zone for ship-ship meet-

ings. This parameter named MDTC is an input for a

model that estimates the probability of ship-ship col-

lision. It takes into account ship dynamics, traffic

patterns and indirectly the human actions (the ma-

neuvering patterns).

504

In the course of the analysis presented in this pa-

per three different charts representing three types of

collision zones were obtained. The statistical analy-

sis shows that the dimension of a collision zone de-

pends mostly on a maneuvering pattern. In case

where both ships perform collision evasive actions,

one chart describes all types of ships analyzed.

However in case where only one ship performs a

collision evasive maneuver, two charts are obtained,

where one considers tankers and another the remain-

ing ship types.

The experiment leading to MDTC chart estima-

tion is based on a ship planar motion model, and the

assumptions concerning deep water and lack of ex-

ternal forces and hydromechanical ship-ship interac-

tions are made. However the size of the vessels un-

der consideration allows the statement, that the sea

conditions prevailing in the Baltic Sea, and especial-

ly in the Gulf of Finland do not affect the results

significantly.

Another important factor affecting the actual

number of modelled accidents is a causation factor.

This topic is not addressed by research presented in

this paper.

ACKNOWLEDGMENT

The authors appreciate the financial contributions of

the following entities: the EU, Baltic Sea Region

(this research was founded by the EfficienSea pro-

ject), the Merenkulun säätiö from Helsinki, the city

of Kotka and the Finnish Ministry of Employment

and the Economy.

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