International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 5

Number 3

September 2011

365

1 INTRODUCTION

The MDTC (Minimum Distance To Collision) mod-

el for ship-ship collision probability estimation is a

geometrical model, with a detailed description given

in the following papers: (Montewka et al. 2010),

(Montewka et al. 2011). In order to provide the

probability of an accident, the model uses the com-

monly adopted approach, which combines a fre-

quency of ship-ship meeting situations given an as-

sumption of blind navigation, and a causation factor,

which quantifies the proportion of cases in which

such a meeting ends up as a collision, due to human

or technical reasons.

The causation factor is a sensitive part of a mod-

el, very much location dependent, thus it is not justi-

fied to use the same value for the different models

(Gluver and Olsen 1998). Applying a causation

probability value derived from a study in another sea

area may save some effort, but then the actual condi-

tions are not addressed at all (Hanninen and Kujala

2009).

Two approaches can be recognized in the litera-

ture in order to estimate the causation factor. The

simplified approach is based on a historical data,

where the causation factor is assumed a ratio be-

tween the registered number of accidents and the es-

timated number of collision candidates (Fujii and

Siobara 1971), (MacDuff 1974), (Inoue and Kawase

2007).

A second approach is more sophisticated, based

on the concept of either event tree (Pedersen 1995),

(Martins and Maturana 2010) or Bayesian Networks

(DNV 2003), (Hanninen and Kujala 2009). This way

of modelling is undoubtedly more time consuming

than the first approach, however it allows getting an

insight into the chain of events leading to an acci-

dent instead of providing just a number.

In order to determine the causation factor for the

MDTC model for three different ship-ship encoun-

tering types (crossing, head-on and overtaking), we

based our study on a modified first approach, which

is relatively quick and straightforward thus robust.

We perform two stage analysis, which combines the

statistical data on maritime accidents and an analysis

of near-collisions based on recorded AIS data.

The causation factor is being defined here as a ra-

tio between the modelled number of collision candi-

dates and the actual number of accidents. However

the available statistics on maritime accidents are not

very detailed, and the type of an accident is not in-

cluded there. Thus there is a need to find a proxy be-

tween a recorded number of accidents and a mod-

elled number of collision candidates (Heinrich et al.

1980), (Inoue and Kawase 2007), (Gucma and

Marcjan 2010).

A Method for Assessing a Causation Factor for

a Geometrical MDTC Model for Ship-Ship

Collision Probability Estimation

J. Montewka*, F. Goerlandt, H. Lammi & P. Kujala

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

Aalto University, School of Engineering, Finland

ABSTRACT: In this paper a comparative method for assessing a causation factor for a geometrical model for

ship-ship collision probability estimation is introduced. The results obtained from the model are compared

with the results of an analysis of near-collisions based on recorded AIS data and then with the historical data

on maritime accidents in the Gulf of Finland.

The causation factor is obtained for three different meeting types, for a chosen location and prevailing traffic

conditions there.

366

It seems justified to analyze the safety of naviga-

tion on the basis of the numbers of both accidents

and near-miss situations. Such a combination of

analyses may better reflect the collision hazard, as

pointed out by (Inoue et al. 2004) and (Inoue and

Kawase 2007).

In air transportation there has been a tendency to

seek out proxy for aviation safety. One commonly

used measure is that of the ”air-miss”, often called a

”near-miss”. According to (Button and Drexler

2006) ”a near-miss involves an aircraft intruding

upon a predetermined safety zone or envelope

around another aircraft”. The reporting procedures

of near-miss in aviation are well founded providing

valuable statistics. In the maritime sector similar

procedures are missing, thus the near-miss can be

detected only by analysis of recorded data and back

propagation of recorded events.

Following this idea, this paper proposes also a

methodology to evaluate the occurrence of near

ship-ship collisions in an open sea area, based on the

AIS data. The method for near-collisions analysis

presented in this work is rooted in a well-established

concept of a ship domain proposed by (Fujii and

Tanaka 1971). An overview of the near collision de-

tection method is then given and applied to the

summer traffic in the Gulf of Finland.

Finally, we compare the results obtained from the

MDTC model, expressed as the number of ”collision

candidates” with the number of near-collisions and

the number of accidents recorded in the chosen area

of the Gulf of Finland. This approach allows us to

quantify the number of modelled ”collision candi-

dates”, with blind navigation assumption behind, to

the number of cases that ended up as close encoun-

ters, where collision evasive actions were taken.

Such quantification is carried out for three major

types of meeting scenario (crossing, head-on, over-

taking). By combining this accurate enough data

with an average annual number of accidents that

happened (which are random, and almost non pre-

dictable), the causation factor for the MDTC model

is obtained.

2 RESEARCH MODEL

2.1 Accident analysis

The annual number of ship-ship collisions in the an-

alyzed location of the Gulf of Finland (the water-

ways junction between Helsinki and Tallinn) is ob-

tained from HELCOM database, that covers a time

period between 1987 and 2007 (Pettersson et al.

2010). During this time, three accidents of this type

took place. Two of them happened during summer

time, and one was related to the ice conditions,

which are out of scope of the analysis presented in

this paper.

According to the aforementioned statistics there

was, on average, one summer collision per ten years.

This assumption is simplified, as the rate of collision

occurence is random, as the first collision happened

in 1996, second in 2001 and between the years 2001

and 2007 no summer collision happened in the area

of investigation. Notwithstanding, we assume that

the annual ship-ship collision frequency in the ana-

lyzed area equals 0.1.

Unfortunately, the database provided by HEL-

COM does not contain any information regarding

type of ship-ship encounter, at which the accident

took place. Thus it is not feasible to compare a mod-

elled number of collision candidates in given en-

counter type (crossing, head-on, overtaking) with an

appropriate number of the accidents. At this point

the results of near- collisions analysis are utilized

and considered a proxy between a model and the

recorded accident data.

Figure 1: The ship domain applied in the near-collision analy-

sis, with the following axes: a = 1.6LOA, b = 4LOA (Wang et

al. 2009)

2.2 Near-collisions analysis

The near-collision analysis applied in this paper is

based on a concept of a ship domain, which accord-

ing to definition given by (Goodwin 1975), is the ar-

ea around the vessel which the navigator would like

to keep free of other vessels, for safety reasons.

Since the first introduction of the ship domain

concept by (Fujii and Tanaka 1971), various re-

searchers have attempted to quantify the size of this

domain. An overwiev of the different proposed do-

mains is given in (Wang et al. 2009). Even though

the ship domain is a well established concept, certain

problems with the application can be identified as

pointed out by (Jingsong et al. 1993). Domains can

be classified by their shape: circular, elliptical and

polygonal domains. A distinction can also be made

between fuzzy domains and crisp domains. Fuzzy

domains such as that proposed by (Pietrzykowski

2008) and (Wang 2010) seem preferable in terms of

367

safety analysis of marine traffic, but are at present

still under development. Crisp domains use a simple

classification of a situation between safe or unsafe,

which evidently is a simplification. Moreover, the

sizes of the domains proposed in the literature vary

quite significantly (Wang et al. 2009).

In this paper, the smallest ship domain found in

the literature, by (Fujii and Tanaka 1971), is applied.

This is justifiable, since the aim of the method pro-

posed in this paper is finding the most critical en-

counters between ships. This domain is defined as an

ellipse with the major axis along the ship’s length

(LOA)and the minor axis perpendicular to the ship’s

beam, as illustrated in Figure 1. The half-length of

the major axis is taken as 4LOA while the half-

length of the minor axis is taken as 1.6LOA. A

number of comments should be made in the use of

this domain:

− the domain is symmetric, which implies that the

possible influence of the COLREGs is not taken

into account;

− another consequence of this symmetry is the fact

that passing behind the stern is considered as

dangerous as passing in front of the bow;

− in the meeting between ships, the largest ship has

the largest domain; this means that for the largest

vessel, the situation is classified as dangerous,

whereas for the smallest vessel, the situation may

still be evaluated as safe;

− the domain is affected by ship length only, neither

ship type nor hydrometeorological conditions are

included in the analysis.

However the latter can be supported by the recent

research, which revealed that the ship domain has a

relatively low correlation with the sea state and wind

force (Kao et al. 2007).

In this section, a brief description of analysis of

AIS data in order to estimate a number of near-

collisions in the selected area of the Gulf of Finland

is given. Recorded AIS data consists of millions of

data points, containing static and dynamic infor-

mation regarding a ship. In order to analyze the mar-

itime traffic in the GOF, this data need to be grouped

into routes. Routes are defined here as a set of trajec-

tories between a departure and arrival harbor as in-

troduced by (Goerlandt and Kujala 2011). The AIS

data is first gathered per ship, based on the MMSI

number. After sorting this data chronologically, the

data per ship is further split up to form individual

ship trajectories, using a methodology described by

(Aarsther and Moan 2009). These trajectories are

then further processed and grouped per route. The

sample rate of these vessel positions in the trajecto-

ries is about 5 minutes on average. In order to enable

a comparison between vessel positions at exactly the

same time instant, the trajectory data is artificially

enhanced to contain data for each second. The ex-

trapolation for the vessel position is performed using

an algorithm suitable for data in the WGS-84 refer-

ence frame following (Vincenty 1975). The ship

speed is linearly interpolated between known values.

It should also be noted that certain vessel types are

not taken into account into the analysis, like tugs are

left out of the analyzed database. This is done be-

cause these vessels are meant to operate in a close

vicinity of merchant vessels. The near collision de-

tection algorithm is shown in Figure 2.

The basic idea is to scan the database for events

where the ship contour of one vessel (i.e. the ship

area in terms of ship length and width) enters the

ship domain of another vessel. If the domain is vio-

lated, the event is labeled as a near collision and rel-

evant details such as time of occurrence, location,

encounter type, ship types and ship flags are stored

for further analysis. The near collision detection al-

gorithm is encoded in MATLAB.

Figure 2: Near collision detection algorithm

368

The algorithm starts with evaluating whether or

not the trajectories of the two considered vessels oc-

cur in an overlapping timeframe. If so, the closest

distance between vessel positions for contemporary

time instances is computed using an algorithm ap-

propriate for geodetic computations according to

(Vincenty 1975). If this closest distance between

points in trajectories is smaller than the extreme val-

ue of the ship domain, the actual vessel contour in

terms of length and width are constructed for the

smaller vessel and the ship domain is constructed for

the larger vessel, for each second. Concurrent ship

domain and a vessel contour are evaluated to overlap

or not. If there is an overlap of a ship domain, the

relevant situational data is stored. If there is no over-

lap, the next case is investigated.

In the analysis of the locations of the near colli-

sions, a distinction is made between three different

encounter situations, as defined in the Collision

Regulations by (Organization 2002). Thus crossing,

head-on and overtaking are considered. Having the

data for the whole Gulf of Finland, we focus on a se-

lected area, which is a crossing of waterways be-

tween Helsinki and Tallinn. The area is bounded by

the following meridians: 024.5deg E and 025deg E

and the parralels: 59deg N - 60deg N.

The results obtained in the course of the analysis,

for the time period analyzed (01.04.2007-

30.10.2007) are depicted graphically in Figure 3.

Figure 3: Results of the near collision analysis

The annual numbers of near collisions, based on

the obtained data, ordered according to the traffic

scenario are shown in Table 1.

Table 1: The annual number of near-collision events in the wa-

terways crossing in the Gulf of Finland.

__________________________________________________

Ships meeting Annual number of near-collisions

__________________________________________________

Crossing 95.0

Head-on 14.0

Overtaking 252.0

Overtaking adjusted 54.5

__________________________________________________

In the yearly perspective the elliptical Fujii do-

main leads to 14 ship domain violations for head-on

encounters and 95 for crossing encounters. Howev-

er, 252 cases are identified for overtaking encoun-

ters. This is due to the fact that the Fujii domain

does not take the regulation of traffic in terms of

traffic separation schemes into account. In order to

get more meaningful results, a heuristic solution for

this is applied, by requiring that the number of do-

main violations for overtaking is equal to the aver-

age number of critical encounters for head-on and

crossing (labeled ”Overtaking adjusted” in Table 1).

To this effect, the Fujii domain is evaluated with a

reduced width of 1.25L_max for overtaking encoun-

ters (as opposed to the original 1.6L_max), where

L_max is the length of the largest vessel in the en-

counter.

2.3 Collision probability modelling

The MDTC model, which is a geometrical model,

estimates a probability of collision between two

ships based on a well founded formula (Kristiansen

2004):

PNP =

(1)

where N

is the number of collision candidates, of-

ten named a geometrical probability of a collision

course and P

is the causation probability, also

called the probability of failing to avoid a collision

when on a collision course. A ship on a collision

course is called a collision candidate, which may end

up as a collision as a result of technical failure or

human error. The causation probability quantifies

the proportion of cases in which a collision candi-

date ends up as a collision.

As a number of collision candidates NA depends

on a number of factors, which are described in the

following part of this chapter, the input data should

be carefully chosen and interpreted before an analy-

sis is carried out. The input values are location de-

pendent, and within a specific location they are very

often also time dependent, for instance:

− an intensity of traffic in the given area (if sched-

uled traffic is observed over the given area, the

intensity of ships will change in the course of the

day),

− a frequency of occurrence of given ship type in

the given area (in general it can be correlated with

369

scheduled traffic, in certain hours more ships of

given type can be expected than in an- other time

spans).

It is also important to observe a correlation be-

tween ship’s main particulars and ship type if sto-

chastic modeling is adopted.

MDTC model applied in this study distinguishes

between three types of ships encounters, these are:

crossing, overtaking and head-on. The probability of

having an accident in case of vessels crossing each

other course, is calculated by means of the following

formula (Endoh 1982), (Montewka et al. 2010):

∑

jiij

gcros

MDTCVE

λλ

sin

]['

sin

(2)

where E’[V

] denotes the expected relative velocity

of all pairs of vessels of types i and j,

denotes the

intensity of the vessels of given type entering the

given waterway, V is the velocity of the vessels ac-

cording to type, and α is the angle of intersection be-

tween the courses of vessels in groups i and j.

In case of parallel meetings, namely overtaking

and head-on meetings, the common formula is used,

and the difference is in a value of intersection an-

gle α. In case of overtaking α <10deg and in case of

reciprocal courses 175deg< α <185deg.

timeparallel

PPTN

(3)

where T

is the overtaking rate (the number of ves-

sels which will overtake another while on parallel

courses, irrespective of the passing distance), P

the probability that the vessels come close to each

other and P

time

is the probability that these two ships

being close to each other will meet in a certain time

period. The latter also reduces the theoretical possi-

bility of ship colliding themselves and is estimated

for scheduled traffic between Helsinki and Tallin.

This probability is not taken into account in case of

E-W traffic, which is more random in nature. The

overtaking rate is obtained by means of the follow-

ing equation (Endoh 1982), (Montewka et al. 2010):

]['

0 ij

T =

(4)

where N is the expected number of vessels in the

waterway on parallel courses, L is the length of wa-

terway, and E’(V

) denotes the expected relative ve-

locity of all pairs of vessels of types i and j. The ex-

pected relative velocity between two vessels is

determined as follows:

[ ]

( )

cos2

jijiij

VVVVVE −+=

(5)

where Vi is the velocity of a vessel of given group, α

means the angle of intersection, which is defined as

the difference between the courses of vessels in

groups i and j.

The probability that the vessels come to a dis-

tance, that results in a collision (P

) is simply esti-

mated as follows:













dPP

(6)

where d is the distance between two ships while

overtaking and B is the breadth of a vessel of a given

class i and j. In order to obtain the results as close as

possible to the results of near-collisions analysis, the

same criteria have to be used. Thus the critical dis-

tance for ships on parallel courses is adopted from

the near-collisions algorithm, and equals 1.25LOA.

Figure 4: The analyzed waterways crossing

Source: (Montewka et al. 2010)

In the course of our analysis we used the AIS da-

ta, which covered a period of seven months of the

year 2007, in which the Gulf of Finland remained

ice free. The analysis presented in this paper consid-

ers specific location in the Gulf of Finland which is

waterways junction between Helsinki and Tallinn.

These water- ways experience dense RoPax traffic

between Finland and Estonia as well as dense traffic

in Traffic Separation Scheme (TSS) heading East

and West.

The ship data-base contains the following data:

MMSI number of ship, time stamp, ship type, ship

length, ship breadth, ship speed, ship course, ship

position (latitude and longitude according to WGS84

reference system). These particulars are used in or-

der to calculate the probability of ship collisions in

the analyzed waterways.

The area under examination and main traffic

streams composition is shown in Figure 4. Four traf-

fic streams are analyzed, according to traffic pattern

observed in the area: North (N), South (S), East (E)

and West (W). The N-S streams consist mostly of

scheduled RoPax ships sailing between Helsinki and

Tallinn while the E-W streams are composed of all

other kinds of ships.

370

Figure 5: Intensities of marine traffic streams

Figure 6: Histograms of main particulars of ships over analyzed area.

371

Maritime traffic in the area is assumed to be a

stochastic process, and is modelled by means of ran-

dom sampling and Monte-Carlo methodology. The

initial traffic database is decomposed into four

smaller databases, according to the four main traffic

streams (Figure 4). Then each stream is modelled

separately, taking into account the non uniform dis-

tribution of ships in time over each stream. The his-

tograms of parameters used in the course of marine

traffic analysis are presented in Figures 5 and 6.

The MDTC value, which acts as an input for the

equation 2, is drawn from the appropriate chart (Fig-

ures: 7, 8, 9). The charts were obtained in the course

of an analysis with the use of a model of ship motion

given the maneuvering pattern and a ship type

(Montewka et al. 2011). The maneuvering pattern,

which decides if both of the ships involved in colli-

sion situation perform collision evasive actions or

only one of them, is chosen randomly with the same

probability of occurrence for each of them (p = 0.5).

Such an assumption may sound simplified, however

there is not enough evidence in the literature to dis-

regard it. In case where the maneuvering pattern one

is chosen, the algorithm checks if there is a tanker

involved, if so then an appropriate MDTC value on-

ly for tankers is chosen.

Figure 7: The obtained MDTC chart for the maneuvering pat-

tern No 1 (Montewka et al. 2011)

Figure 8: The obtained MDTC chart for tankers - the maneu-

vering pattern No 1 (Montewka et al. 2011)

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

tern No 2 (Montewka et al. 2011)

The annual number of collision candidates ob-

tained with hte use of MDTC model is presented in

Table 2. The results are divided into three meeting

scenarios (crossing, head-on and overtaking). Within

these scenarios there are different sub-scenarios

which represents meetings of ships sailing in various

streams.

Table 2: The annual number of collision candidates in the wa-

terways crossing in the Gulf of Finland, obtained by means of

MDTC model (the avarage values).

___________________________________________________

Ships meeting Annual number of collision candidates

___________________________________________________

Crossing 5538

___________________________________________________

Head-on N-S 1.0

Head-on E-W 57.0

Head-on All 58.0

___________________________________________________

Overtaking N-N 164

Overtaking S-S 156

Overtaking E-E 28

Overtaking W-W 34

Overtaking All 382

___________________________________________________

3 RESULTS

In the course of presented analyses we obtain a data

regarding near-collisions, number of accidents and

modelled number of collision candidates for the spe-

cific location in the Gulf of Finland.

The aim of this reasearch is to develope a causa-

tion factor for the MDTC model by means of a com-

parative study. The values of the causation factor

) are strongly location dependent, as the original

studies regarding this parameters have been con-

ducted in the specific locations (eg. straits in Japan,

the Dover Strait) it is difficult to assess how the re-

sults obtained there can be transferable to other sea

areas. The P

value is also highly dependent on a

geometrical model used for the probability of ship

accident estimation, thus transferring the same value

between different models seems not justified from

the scientific point of view.

372

In our approach we estimate the causation factor

that is related to the MDTC model, based on the fol-

lowing formula:

( )

∑

−

collnear

SHFmP

(7)

)(

mcandcoll

mcollnear

SHF

−

(8)

where SHF is a ship handling factor defined for each

type of meeting m individually (m = [head-on, over-

taking, crossing]), for the specific value of S H F see

Table 3, N

near-coll

is a number of observed near-

collisions, N

coll-cand

is a number of modelled collision

candidates and N

is a number of recorded accidents.

As a result the causation factors for three types of

ship/ship encounter were estimated (Table 4).

Ship handling factors presented in Table 3 gov-

erns a ship handling process, showing a difference

between blind navigation model and real traffic for

different encounter types.

Table 3: The ship handling factor (SHF) for three types of ship-

ship encounter, for a specific location in the Gulf of Finland.

___________________________________________________

Type of meeting The SHF for given events

___________________________________________________

Crossing 1.7 * 10e - 2

Head-on 2.4 * 10e - 1

Overtaking 1.4 * 10e - 1

___________________________________________________

In Table 4 the values of the causation factor for

the MDTC model are gathered. However further re-

search which would cover the greater sea area lead-

ing towards a better definition of the causation factor

should be carried out.

The numbers for the causation factor proposed

here consider a specific geometrical model (MDTC),

ordered traffic with waterways crossing, continous

surveillance from VTS stations, presence of Traffic

Separation Schemes and an intense RoPax cross traf-

fic. The proposed causation factors make a distinc-

tion between type of ship-ship encounter. The model

is applicable only for the ”summer traffic”, which

means, that presence of ice is not considered.

Table 4: The causation factor for the MDTC model for three

types of ship-ship encounter.

___________________________________________________

Type of meeting The causation factor

___________________________________________________

Crossing 1.04 * 10e - 5

Head-on 1.46 * 10e - 4

Overtaking 0.85 * 10e - 4

___________________________________________________

The general relations between analyzed types of

event (modelled number of collision candidates, ob-

served number of near-collisions and recorded num-

ber of accidents) for the analyzed location are de-

picted in Figure 10.

Figure 10: The general relations between each type of event

4 CONCLUSIONS

This paper addresses a problem of defining the cau-

sation factor for a given geometrical model. We pro-

pose a straightforward methodology, which is based

on recorded near-collisions (obtained in the course

of AIS data analysis) and actual collisions (obtained

from HELCOM accidents database). The method

estabilishes the ratios between the recorded number

of accidents, the recorded number of the near-

collisions and the modelled number of the collision

candidates. Knowing these values, it is possible to

define a causation factor that constitutes a link be-

tween a geometric model for ship-ship collision fre-

quency estimation and a number of accidents due to

the given parameters of marine traffic and surround-

ings.

Making a comparative study we defined the cau-

sation factors for the MDTC model, for three ship-

ship encounter types. The estimated values of causa-

tion factors for the selected area of the Gulf of Fin-

land and given types of vessels sailing there are of

the following orders of magnitude: 10e - 5 for ships

crossing and 10e - 4 for ships meeting each other on

parallel courses.

Although the methodology behind this analysis is

straightforward, the results are promising, however

there is a need for more extensive analysis, that

would cover a larger sea area.

ACKNOWLEDGMENT

The authors appreciate the financial contributions of

the following entities: the EU, Baltic Sea Region

(this research was founded by EfficienSea project),

Merenkulun säätiö from Helsinki, the city of Kotka

and the Finnish Ministry of Employment and the

Economy.

373

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