PROBABILISTIC MODELS FOR SHIP SAFETY
ASSESSMENT IN RESTRICTED AREAS BASED
ON UNIFORM AND NORMAL DISTRIBUTION
JOINTS
Gucma L.
Maritime University of Szczecin, Szczecin, Poland
Abstract: This article presents selected probabilistic models used for the description of ship’s
po-sition in restricted areas. Two models based on a joint of normal and uniform distributions have been discussed in detail. These models can be applied for the estimation of accident probability and manoeuvring risk taken by ships sailing in restricted areas.
1. General character of random variables used for the determination
of a probability of ship’s crossing safe fairway boundaries
This quantitative analysis of safety will deal with variables describing the process of ship movement in a restricted area. These variables can be considered as random variables and described by means of probability distributions. This paper presents the construction of two selected distributions of random variables used for the determination of the probabil-ity of ship’s collision with stationary obstructions.
In most applications of marine traffic engineering, where a ship proceeds along a preset route with the coordinates y = 0, the distribution of ship position distances from the fair-way centre line can be transformed into a conditional distribution satisfying the condition x1 < X < x2, where x1,x2 make up the examined fairway section. The corresponding
distri-bution function has this form:
) x X x | y Y ( P ) x , y ( FY|X 1 2 (1) This kind of distribution can be easily used for the determination of the probability of ship’s collision with a shore or port structures at a point xi as (Fig. 1):
)
D
(
F
1
P
i i x X | Y i Ax
(2) where:Di - available width of the manoeuvring area at a point i.
Attempts to systematize and build probabilistic models related to the safety of ships navi-gating in restricted areas were made by [Guziewicz, 1996] in his doctoral dissertation. He searched for probabilistic models of maximum distances of ship’s extreme points to the fairway centre line and models of the energy of ship’s impact on the berth. He indicated that the normal distribution is suitable for the description of the former group of variables, while the gamma distribution is good for the latter. In his work Guziewicz also introduced the methodology of constructing general probabilistic models used in marine traffic engi-neering.
Fig. 1. The method of constructing a ship’s safe manoeuvring area and the determination of ship-shore collision probability
2. The distribution of ship’s centre of gravity position based on the
normal distribution
The ship moving along a preset route (e.g. the fairway centre line), shifts off that line on a random basis. Ship’s positions are usually understood as the positions of its center of gravity. The distribution of the distance of ship’s gravity centre from the fairway centre line is fit for the examination of safety in systems where ship’s dimensions can be ne-glected in relation to the area dimensions. In other cases, i.e. in most restricted areas, it is impossible to avoid simplifications and inaccuracies resulting from failing to know the ex-act position of the ship relative to the area (ship’s heading and charex-acteristic points of its waterline). Simplifying the considerations, we have to shift the distribution to the port and starboard by a certain value equal to the half of ship’s breadth (assuming there is no drift
angle) or to the ship’s breadth half increased by the value resulting from ship’s drift angle (Fig. 2). With the assumption that the distribution of ship’s gravity center distances from the fairway centre line is normal, the shifted distribution for the port side maximum dis-tances from the fairway center line obtains this form:
2 sc 2 2 ) B y sc m y ( sc B sc l
e
2
1
)
y
y
(
d
)
y
(
d
(3) where:msc,sc - mean and standard deviation of ship’s gravity centre distance from the fairway
center line from the sample;
yB - value of the distribution shift equal to yB = B when drift angle in not considered or yB =
Bcos + 0.5 Lsin when it is considered and the waterline is assumed to be rectangular.
Fig. 2. The distribution of ship’s gravity center position and the problem of the determination of the extreme points shift
In the research covering offshore areas when the ship is treated as a material point the fol-lowing distributions can be used as a model of ship’s gravity centre position in relation to a preset route:
normal [Gucma L., 1999a; Guziewicz, 1994 and 1999; Iribarren, 1999], uniform [Purcz, 1998],
joint normal and uniform distribution, extreme value,
In spite of widespread research in quest of general statistic models of ship’s position relative to the fairway, the type of distribution of ship’s gravity centre positions is strictly de -pendent on the area shape, type of manoeuvre performed, ship type and hydro-meteoro-logical conditions. The commonly adopted procedure, therefore is the use of empirical distributions which are estimated experimentally [Gucma L., 1999b].
3. The joint of normal and uniform distribution
For probabilistic modeling of ship’s position in narrow fairways is seems that a joint nor-mal and uniform distribution is closer to reality than a uniform or nornor-mal distribution used separately. Assuming that the density functions of these distributions have this form:
2 2 ) m x ( e 2 1 ) x ( f (5) and b x a dla a b 1 ) y ( f (6) we can compute the joint distribution by means of this relationship:
dx ) x z ( f ) x ( f dz ) z ( dF ) z ( f x y (7)
Introducing a new variable u(xzm)/ we get:
du
e
2
1
a
b
1
dx
e
2
1
a
b
1
)
z
(
f
(b z m)/ / ) m z a ( 2 2 u b a 2 2 2 ) m x z (
(8) that is: F (b z m) F (a z m) a b 1 ) z ( f nz nz (9)where Fnz is a distribution function of standardized normal distribution having this form:
du e 2 1 ) z ( F z 2 2 u nz (10)
It should ne noted that if b a , then the value of one of the distribution functions in relationship (9) equals 1 or 0. The resultant distribution is shown in Figure 3. It can be used for modeling ship’s positions in restricted fairways. Looking closely at its shape one can notice that ship’s positions around the fairway center line are similar to the uniform distribution – the navigator does not attempt to bring the ship back to the centre line of the fairway within a certain range from the centre (this can be due to the fact the navigator is unaware of the real position of the ship). In a situation when the ship approaches the safety boundaries of the fairway, the navigator takes action to prevent an accident by al-tering the ship’s course.
Fig. 3. Joint normal and uniform distribution, for these parameters: a = 10, b = 10, m = 0, = +2
4. The distribution composed of uniform and normal distributions
In offshore areas a theoretical model of ship’s positions often makes use of a distribution that combines normal and uniform distributions [Fuji, 1977]. Such a distribution is very convenient for computations as well as for modeling the phenomenon. This model as-sumes that certain percentage of ships that represent random movement is subject to the uniform distribution in the adopted limits, and the remaining part representing movement along a preset route is subject to the normal distribution. Although many authors quote Fuji’s works [Gluver and Olsen, 1998], none including Fuji himself gives the form of such a distribution. Presented below is the method of defining this kind of distribution. This is a distribution consisting of the normal and uniform one. It can be built assuming that n per cent of the movement is subject to normal distribution, while (n 1) to uniform distribution. The condition of distribution standardization is satisfied by establishing the boundaries of the truncated normal distribution from –b to b so that:
1
)
n
1
(
n
dx
a
2
n
1
dx
e
2
1
a a b b 2 2 ) m x (
(11) The boundaries from –b to b are established by this relationship:n 1 be 2 dx e dx e 2 1 2 2 ) m b ( b 2 2 ) m x ( b 2 2 ) m x ( (12) using the normal distribution truncated on both sides, satisfying the condition
b bn
dx
)
x
(
f
. The value b can be determined by numerical methods. The limits of uniform distribution –a to a are fixed depending on the area in which random movement occurs. Assuming that m = 0 and a >> b, we obtain a case the most common in practice in which it is assumed that random ship movement (uniform distribution) takes place in a much larger area than the movement along a preset route (normal distribution). The con-struction of the distribution consists in cutting the ends of the normal distribution (Fig. 4). The mixed distribution has the following density function:
a
y
b
for
a
2
n
1
b
y
b
for
a
2
n
1
e
2
1
b
y
a
for
a
2
n
1
a
y
i
a
y
for
0
)y
(f
2 ) / y ( (13)This distribution, as can be observed from an analysis of one of its density functions (Fig. 4), at its ends assumes a form close to the uniform distribution, while in the middle it takes a form of the normal distribution. Therefore, it is suitable for modeling the positions of two groups of ships, of which one proceeds along a preset route, the other does not. This author also proposes to use this distribution in restricted port areas and fairways. This is justified in the case of a symmetrical fairway restricted between a and a two groups of ships exist: limited by the draft (large ships) and unlimited (small ships, barges etc.). The latter ships move fitting the uniform distribution, keeping the ship within the fairway
limits, while the former ships, moving according to the normal distribution, try to keep to the fairway center line as close as possible.
Fig. 4. A mixture of uniform and normal distribution for a specific n
5. Conclusion
The paper presents the construction of two major distributions used for the modelling of ship’s position in relation to the water area. These distributions have been built as a joint and mixture of the uniform and normal distributions. They can be used in solving various problems of marine traffic engineering, particularly those concerning risk assessment and accident modeling.
References
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