More than 30 years have passed since Macduff and Fujii expressed their first ideas about the modeling of ship grounding. The probability of ship grounding is usually calculated by multiplying two probabilities named geometrical and causation probabilities. Geometrical probability gives the probability of a ship being a grounding candidate, which means a ship that will run aground if no evasive action is performed. Consequently, the causation probability will give the probability of a grounding candidate not to do any evasive action and then goes aground.
Many geometrical models have been presented during these years for estimating the probability of grounding. However, after all these years, there is still lack of a well-defined geometrical model for analyzing the probability of ship grounding.
This paper represents four of the most cited geometrical probability models (the models of T. Macduff, Y. Fujii, P.T. Pedersen and B.C. Simonsen) and discusses about their weaknesses and strengths. In the third part of the paper, some improvement for Macduff and Fujii's models are suggested. In the fourth part of the paper, the capability and sensitivity of all four models are assessed by calculating the probability of ship grounding with real traffic information from the AIS (Automatic Identification System) data of the Gulf of Finland in 2008.
Keywords: Ship Grounding, Ship Stranding, Geometrical Modeling, Traffic Density, Traffic Volume, Fuzzy Ship Domain.
Introduction
Ship grounding accounts for about one-third of commercial ship accidents (Jebsen and Papakonstantinou 1997; Kite-Powell et al. 1999). To give few examples, about 20% of all tanker losses between 1987 and 1991 were due to grounding (Amrozowicz et al. 1997). Zhu et al. [1] [cited in (Samuelides et al. 2009)] have reported that the total losses of all ships during the period 1995-1998 were 674 in numbers and 3.26 M in Gross Tonnage (GT), where 17% in number and 24% in GT were due to grounding. Also more than half of all the accidents in the Gulf of Finland are grounding accidents (Kujala et al. 2009). Moreover, 47% of all accidents of the Greek ships larger than 100 GT all over the world between years 1992-2005 were reported as grounding (Samuelides et al. 2009).
Nowadays most of the risk assessments on ship grounding are done by the help of Macduff's (Macduff 1974) and Fujii's (Fujii et al. 1974) first ideas. They both used the concept of multiplying the number of ships being on a grounding course (grounding candidate) and the probability of not making evasive maneuvers (causation probability) to calculate the number or the probability of grounding accidents.
Grounding candidates are those ships that go aground if nothing, internally or externally, changes; it means that if nobody onboard does any evasive actions or the environmental situation does not change [2] . Some models give grounding candidate probability whereas some others yield the number of grounding candidates. The number of grounding candidates is the number of the ships that are grounding candidates through the analysis area; and if this number is multiplied by the causation probability, it will yield the number of groundings. Grounding candidate probability is the probability that a ship is a grounding candidate during one passage through the analysis area; and if this probability is multiplied by the number of vessels in the traffic, it will yield the number of grounding candidates.
Causation probability informs how probable it would be that the situation (internally or externally) does not change in favor of the ship, in different given scenarios.
Therefore, according to present knowledge, for finding the probability of grounding in a given location or scenario, it is needed to have both the number of grounding candidates and causation probability. There are different internal and external factors that affect both probabilities. Internal factors are those that are related to the ship, herself; and external factors are those that will appear depending on the environmental situation related to the location of the ship.
Different factors like human factors, vessel and route characteristics, atmospheric and situational factors should be statistically analyzed to get the clear contribution of each specific factor affecting the causation probability (Mazaheri 2009). However, the grounding candidate probability can be obtained via so-called geometrical model. Although the contributing factors in geometrical modeling and also the level of models' precision is still a matter of question (Mazaheri 2009), there are plenty of useful geometrical models available in the literature.
This paper includes three main parts. In the first part the most cited geometrical models in the area of grounding probability (models of T. Macduff, Y. Fujii, P.T. Pedersen and B.C. Simonsen) have been represented and their strengths and weaknesses are discussed. In the second part, some improvements are suggested for the models of Macduff and Fujii. In the last part the discussed models have been used for analyzing the grounding probability in a real case in the Gulf of Finland (near Sköldvik). At the end, the conclusion is presented.
Dominant Models
The existing models in the literature can be divided into two groups as analytical and statistical models (Mazaheri 2009). Analytical models include Fujii's and Macduff's models together with those who have followed them in their modeling like (Fowler and Sørgård 2000) and (Kristiansen 2005). None of them use ship traffic distribution, which is the probability density function of the lateral position of ships on a waterway (see Figure 3). On the other hand, the models of Pedersen and his followers like (Simonsen 1997), (Karlsson et al. 1998), (Otto et al. 2002), (Gucma 2006) and (Quy et al. 2007) can be called statistical, as the ship traffic distribution has been used in the models.
On the other hand, the existing models deal with two different major scenarios as stranding and grounding. (Kristiansen 2005; Mazaheri 2009) "Stranding is the event that a ship impacts the shore line and strands on the beach or coast. It happens when the track of the ship intersects the shoreline by either navigational error or drifting. However grounding is the event that the bottom of a ship hits the seabed. It happens when a ship is navigated through an individual shoal in a fairway while her draft exceeds the depth." (Mazaheri 2009)
Analytical Models
(Macduff 1974)
Macduff argued that the real probability of grounding (PRG) would be the product of geometrical probability (PG) and causation probability (PC):
( 0 )
Then, by the help of Buffon's needle problem, he calculated the geometrical probability of random grounding (assuming random navigation through the channel) as:
( 0 )
where:
s is the track length of the ship or stopping distance
C is the width of the channel or waterway
Figure 1: Probability of hitting the wall of the channel according to (Macduff 1974).
Macduff's model does not consider any shoal in the middle of the channel, so it just yields the probability of stranding. A matter for consideration in Macduff's model is that the width of the channel should be assumed not to change significantly when moving along the channel. Since the number of natural channels that meet this criterion is, if not say zero, very few, the scope of the formula will be limited to just some manmade channels. In addition Macduff's model does not take into consideration the length of the studied waterway, which is questionable as logically ships have more possibilities to run aground in a longer channel than in a shorter one.
Another considerable point about Macduff's model is that he has used Buffon's needle problem, which is a 2D model, to find a geometrical probability of stranding (hitting walls of a channel). In his model, the ship has been considered as a needle or a line, which is one dimensional. When applying this approach, two matters need to be considered. First, vessels (unlike needles) are not dropped into the channel from the sky, so their positions and headings do not fit into the uniform distribution (Kunkel 2009). Also one may argue that the path of the ship (unlike the needle) is not straight either. However it should be noticed that a straight line from the starting point to the end point of the path is enough as a criterion. The second matter is that since needles (as line), contrary to ships, have one dimension, it could be understood that the breadth and the draft of the vessel and the depth of the channel have not been taken into the attention. However, it can be argued that hitting the wall of a channel, itself, means the draft has exceeded the depth of the channel; and therefore it can be understood that the draft and the depth has been taken into account.
(Fujii et al. 1974)
Fujii's model together with Macduff's was one of the earliest geometrical models designed for grounding risk modeling, and most of the researches in this area have been done based on their works.
Fujii has argued that the approximate number of ships going aground in a waterway would be:
( 0 )
where:
V is the average speed of the traffic flow
Ï is the average density of the traffic flow
D is the linear cross-section of the obstacle shallower than the [average] draft
B is the [average] ship width
D+B is the effective width of the obstacle or shoal
PC is the probability of mismaneuvering or causation probability
Fujii brought into attention that since D is usually much larger than B, B can be ignored.
( 0 )
Also he has mentioned that when D is much larger than W, width of the route, the formula can be rewritten as:
( 0 )
where:
Q is the traffic volume and is equal to ÏWV
W is the width of the channel
However, the question is how possibly D could be larger than W? The only possibility is when the ship omits a turn in vicinity of a shoal and grounds on the shoal; otherwise D could be maximally equal to W and in this case all ships would be grounding candidates.
Although Fujii has showed that he has taken the draft into consideration by mentioning that "… obstacle shallower than the draft", like Macduff's model, the channel's depth and the vessels' drafts have not been considered directly for calculating the grounding candidates. In addition, Fujii's grounding model is similar to his model on collision with fixed object (Fujii 1983). Therefore, contrary to Macduff, he has not considered the probability of hitting the wall of the waterway, and his model just yields the probability of grounding and not stranding. On the other hand, Fujii, like Macduff, did not take the length of the waterway into account.
Statistical Models
(Pedersen 1995; Simonsen 1997)
Pedersen's model is the most used geometrical grounding model in recent years. Simonsen's model is actually a revised version of Pedersen's work. Their models have been used in two recent risk analysis software [GRACAT (Hansen et al. 2000; Hansen and Simonsen 2001) and GRISK (Ravn et al. 2008), currently called as IWRAP Mk2 (2009)] for analyzing grounding and collision probabilities.
Pedersen has defined an imaginary route with a bend in the navigation route around an area where ships with a draft above a certain level may ground (Figure 2). Again, does it mean that he has considered the ship's draft and the depth of the channel in his model? Like what has been posed, do we really need to think about how these factors (draft and depth) can affect the model; or going aground, by its own, means that those factors have been considered into the model?
Pedersen and Simonsen have categorized the grounding scenarios into 4 different categories (Pedersen 1995; Simonsen 1997), and the estimated frequencies of grounding on the shoal can be obtained as a sum of the four different accident categories. The third and forth categories are about grounding due to evasive maneuvers and drifting ships; and they are not represented here in this paper.
In category I, ships follow the ordinary and direct route at normal speed. The accidents are due to human error and unexpected problems with the propulsion/steering system which occur in the vicinity of a shoal. The simplified expression of this category, according to Pedersen is:
( 0 )
and according to Simonsen is:
( 0 )
The simplified expression for the category of ships which fail to change course at a given turning point near the obstacle (category II), according to Pedersen is:
( 0 )
and according to Simonsen is:
( 0 )
where:
FCat is expected number of groundings per year
i is the index for ship class, categorized by vessel type and dead weight or length
Pci is the causation probability, i.e. ratio between [actual] ship groundings and ships on a grounding course
Qi is the number of movements per year of vessel class (i) in the considered lane
L is the total width of the considered area perpendicular to the ship traffic
Bi is the collision indication function, which is one when the ship strikes the structure or shoal and zero when the candidate colliding ship does not hit the obstacle, that is, passes safely or grounds prior to collision or grounding on the considered shoal.
P0 is the probability of omission to check the position of the ship
d is the distance from obstacle to the bend in the navigation route, varying with the lateral position of the ship
ai is the average length between position checks by the navigator
z is the coordinate in the direction perpendicular to the route
(zmin,zmax) are the transverse coordinates for an obstacle
fi or fi(z) is ship track distribution
Figure 2: Distribution of ship traffic on a navigation route [source (Pedersen 1995)].
As is seen, Simonsen has replaced the Pedersen's B factor by integration boundaries ZMax and ZMin. Either B factor or integration boundaries let us to consider those ships that are in grounding or collision courses only (Figure 3).
Figure 3: Grounding candidates for ships on straight route [adapted (Rambøll 2006)].
Pedersen's and Simonsen's models include the parameter ai, average distance between position checks by the navigator. The parameter depends on navigational environment and ship characteristics as type and size. It should be estimated separately for every location and ship group. To get indicative results, some global values can be used. Nonetheless, the mentions of numerical values of ai are very rare in the literature.
Simonsen has assumed the event of checking the position of the ship is a Poisson process; thus he replaced P0 in Pedersen's equation [Eq.( 0 )] by an exponential function [Eq.( 0 )]. This has made Simonsen's model less sensitive to the values of ai, which will be shown later. Simonsen has mentioned that the theoretical result achieved by the model is quite sensitive to both the causation probability, PC, and the distance between each position checking, ai.
The most important advantage of their models is that instead of traffic volume and traffic densities, which have still some vagueness in their definitions, Pedersen and Simonsen used traffic distribution. Since the traffic distribution also shows the location of the vessels in the waterway, it is more precise than traffic volume or density for estimating the grounding candidates. In addition, it has made their models suitable for analyzing both grounding and stranding accidents. However, the main issue in using the traffic distribution is that in practice it needs a complete AIS database about the traffic to be precise; otherwise it is just traffic estimation, which decreases the accuracy of the method.
Improvements
Suggested Improvements for Macduff's Model
The probability should always be less than or equal to 1, and it is so for geometrical probability of grounding (PG). Therefore it is obtained from Eq. ( 0 ) of Macduff's model that ; which is not always the case.
By reconsidering the Macduff's first idea about using Buffon's problem and also by applying the Buffon's needle problem's criteria differently, the authors suggest the geometrical probability of hitting the wall of a channel (stranding) to be estimated by Eq. ( 0 )4 [3] :
( 0 ) [4]
The Buffon's problem is defined and solved for a needle laying down on just one line, while for grounding it should be solved for two parallel lines. Also it is not important if the needle intersects the line with its tail or head, while the direction of the hitting is another important matter for grounding event. Nevertheless, by assuming that ships always move forward (hitting with bow) and also in a channel (two parallel lines), the Buffon's equations [Eq.( 0 )4] still can be applied for grounding event without any changes.
According to Figure 4, which is based on Eq.( 0 )4, PG is almost equal to 1 for . Since Macduff has mentioned that "it is estimated that ships are capable of stopping within a distance equal to 20 times their length", s is equal to 20L, where L is the length of the ship. Therefore by accepting Macduff's definition, ships are certainly grounding candidates for cases that .
Figure 4: Probability of Grounding v.s.
As has been mentioned, the geometrical model should present the candidate ships for grounding. In other words, it should present the probability of ships running aground while they are navigating in blind situation. Blind situation or blind navigation means not to do any action to avoid grounding; or in other words, not to adjust the course and speed over ground during the voyage in the channel. If it is so, does the stopping distance (s) has any meaning in blind navigation? The ship is supposed not to do any evasive action to avoid grounding; while stopping distance means that somebody on board has did some efforts to stop the ship before going aground. Moreover, the grounding can be avoided not only by stopping but also by changing the course. For instance, in high speed (usually more than 12 knots) the accident can be avoided more readily by turning than by stopping (Mandel 1967).
On the other hand, navigating under blind situation cannot be well defined; always there would be the risk of going aground in blind navigation, even in safe routes. For instance in Figure 5, there would always be risk of grounding for a ship navigating under blind situation, even for ships navigating between two region 1s (safe transit channel). However, in reality those ships that have entered the region 1 are those who are candidates for going aground; if they do evasive action, they will survive. Still for some reasons with different probabilities, all of them will not be able to avoid entering region 2. Most probably those ships that have entered region 2 will run aground, because the maneuvering ability of the ship to avoid grounding will be decreased (Amrozowicz et al. 1997). The widths of the regions 1 and 2 are dependent on different parameters such as vessel's characteristics and environmental conditions.
Macduff's model can be modified by the help of Figure 5. If C is considered as the width of the safe transit channel (Figure 5), then Macduff's PG would be the probability of being a grounding candidate. In this case a new definition for s is needed, which with the help of Simonsen idea (assuming as a Poisson process) can be defines as:
( 0 )
where:
V is the (average) speed of the ship(s)
a is the (average) time between each position checking by the navigator(s)
L is the (average) length of the ship(s)
C is the (average) width of the channel
Figure 5: Hypothetical waterway [adapted (Amrozowicz et al. 1997)].
Since the probability in Macduff's improved model is very sensitive to the changes of where (Figure 4), it is very sensitive to any changes of s. This shows the need of having a reasonable definition for s. A question still remains whether s should be the stopping distance or something else [like Eq. ( 0 )]?
A better method for describing the situation (for both grounding and stranding accidents) could be ship domain, as demonstrated in Figure 6. If the vessel does not turn before the region 1 of the domain hits the channel's wall or a shoal, the vessel will be a grounding candidate. However, she still can survive if an evasive action takes place. If the evasive action is not performed and the region 2 of the domain hits the wall or shoal, most probably the ship will go aground.
According to (Zhu et al. 2001) the ship domain should be defined according to the navigator's sense of safety. The fact is that in the case of grounding, the safe area cannot be separated from the dangerous area by a simple line; so that passing the line would mean the ship is in danger, and not passing would mean she is safe. Thus, to improve the above idea, an interesting tool could be fuzzy theory by Zadeh [5] . By applying fuzzy logic method on different affecting factors, the above theory can be modified in order to give a better estimation of grounding probability. This method can be applied for every single factor affecting the size and the shape of the ship domain. The resulted domain is fuzzy ship domain (Figure 7). Using fuzzy boundary for ship domain is proposed by (Zhao et al. 1993) and is developed by (Pietrzykowski and Uriasz 2009) for ship-ship collision; however it has never been used for ship grounding.
Figure 6: An example of hypothetical ship domain for grounding.
Figure 7: An example of fuzzy ship domain for grounding.
The main issue here is to define and calculate the shape and the size of the domain's regions. They are related to many factors such as size and the speed of the vessel and topography of the sea bed. Also a suitable ship domain for grounding is a 3D domain; and a 2D domain does not work properly for grounding.
The fuzzy ship domain is suitable for both geometrical probability modeling and also for real time grounding probability analysis of an individual ship, which could be useful for Decision Support Systems (DSS) on board the ships or at VTS (Vessel Traffic Service) centers [6] [Cited in (Pietrzykowski 2002)]. However, the geometrical models are just practical for analyzing the whole traffic in a specific area, which is useful for predicting future risks and precautionary plans.
Comments on Fujii's Model
Referring to Eq. ( 0 ), since the number of grounding (N) has an inherent time factor [7] , its dimension is . By this definition, the dimension of the density of the traffic flow according to (Fujii et al. 1974) would be gained as:
( 0 )
if:
[N] [8] = Number of the ships / T
[D] and [B] = L
[V] = L / T
[PC] = Dimensionless
If it is so, referring to Eq.( 0 ), the dimension of traffic volume (Q) would be gained as:
( 0 )
Fujii has mentioned that "the traffic flow density is equal to the traffic volume per unit width of waterway" (Fujii et al. 1974), which makes the authors to reach to the conclusion that the traffic volume (Q) should be equal to the product of the traffic flow density (Ï) and width of the waterway (W):
( 0 )
On the other hand, Fujii has mentioned another definition for traffic volume in his paper as "the product of the traffic density and the average speed" (Fujii et al. 1974). Therefore:
( 0 )
As is showed, three different dimensions for traffic volume could been extracted from his paper, while he has actually mentioned the dimension for traffic volume as "the number of the ships per km2" (Fujii et al. 1974) or
( 0 )
which does not match none of them above, and also is similar to what is extracted for traffic density. As is seen, there is no clear definition neither for traffic volume nor for traffic density according to (Fujii et al. 1974).
Suggested Improvements for Fujii's Model
In authors' opinion, the traffic density should be defined as the number of ships per unit area of the waterway. However, there are two points that should be considered:
Since the traffic is compressible and is not stationary, the time is an inherent factor for traffic density. Thus the time window has to always be considered when defining the traffic density. To overcome this barrier, the length of the area, within which the traffic density is going to be defined, will be connected to the desired time window. It means that the length of the studied area is equal to the distance that the traffic (ships) travels within the desired time window.
Since traffic of, for instance two small boats differs from traffic of two VLCCs (Very Large Crude Carrier), the dimensions of the vessels also should be considered in traffic density definition. Therefore a dimensionless parameter, size factor, should affect the traffic density.
Thus the authors recommend defining the traffic density as:
( 0 )
where
L* is the distance that traffic travels within desired time window
As a result the dimension of the traffic density would be the same as what is extracted from (Fujii et al. 1974) and then, with the help of mass flow rate concept in fluid dynamic, the traffic flow rate could be defined as:
( 0 )
where V and W are the average velocity of the vessels and the width of the waterway, respectively.
If it is so, the dimension for the traffic flow rate would be gained as:
( 0 )
(Kristiansen 2005) named this traffic flow rate (Q) as the "arrival frequency of meeting ships", when he was analyzing the expected number of head-on collisions in his book.
General believe about the traffic volume is the number of the vehicles passing an imaginary line during a specific period of time (Jacobson 2007). Therefore, regarding to Eq.( 0 ) the traffic flow rate cannot be taken as traffic volume. However, can traffic volume be compared with a fluid dynamic concept so-called flux? If it is so, traffic volume (or traffic flux) could be defined as the number of vessels navigating through a unit line per unit time; or the product of the traffic density and the average speed [Eq.( 0 )], which is similar to one of the Fujii's definitions for traffic volume.
( 0 )
Since the used density in this formula has been redefined by Eq.( 0 ), the size of the vessels also affects the traffic volume.
In this regard, the traffic flow of ships towards a shoal with the effective width of D+B and average speed of V can be defined as:
( 0 )
where:
T is the time window in which the traffic flow is desired to be calculated
Thus, the number of grounding candidates per time unit would be calculated as:
( 0 )
As a result, the number of groundings (N) per time unit could be calculated by using the probability of mismaneuvering (causation probability):
( 0 )
As is seen, the result is similar to what Fujii presented in his paper, but with changes in the definitions of traffic density and traffic volume. Without those changes, the yielded annual number of groundings would be different if different time windows were used.
It should be borne in mind that the traffic density has been considered constant during the time window for calculating the traffic flow. If the average traffic density is not used, or the time window is not small enough so that the traffic density could be considered constant in it, then traffic density (Ï) should be defined as a function of time:
( 0 )
The question is that how easily the traffic density can be defined as a function of time? There are many factors that should be considered, like the season, day time, weather condition, economic situation, and some of them are hard to predict. However, even if the traffic density can be defined as a function of time, still using traffic density in geometrical modeling of grounding means that the traffic has been considered uniformly distributed. Thus, it may unnecessarily decrease the accuracy of the yielded results. This could be considered as the main disadvantage of using traffic density (analytical models) instead of actual ship traffic distribution (statistical models) in geometrical modeling of grounding.
Application of Dominant Models
For calculation of grounding frequency, a waterway on the way to Sköldvik (Figure 8) has been chosen. Two locations have been indicated from the area for two different types of calculations. First type of calculation is about Pedersen's category I of groundings; and the chosen location for this purpose is the waterway between islands on the way to Sköldvik (Location 1 in Figure 8, Figure 9). For this type of calculation, the models of Macduff, Fujii and Pedersen are used. It should be noted that for type I calculation the models of Pedersen and Simonsen do not differ. The second type is about Pedersen's category II of groundings (Location 2 in Figure 8, Figure 12). For the second type of calculation, just the models of Pedersen and Simonsen are used, as other models do not consider the scenario of omitting a turn in vicinity of a shoal.
Figure 8: Waterway on the way to Sköldvik.
Majority of vessels navigating to Sköldvik port are tankers, so only tankers are taken into account in the calculations. AIS data of year 2008 is used to get traffic characteristics as tanker length and draft.
All the calculations yield either the number of grounding candidates or the grounding candidate probability. It means that the causation probability does not affect the calculations or in other words, PC has been considered as equal to 1.
Type I Calculations
The number of tankers that navigated northward along the waterway, for the location 1, was 1176 in 2008. The longest tankers on the waterway were 277 m long. The average length of tankers was approximately 147 m and the average breadth was 22.9 m. The studied channel was approximately 900 m wide and 2000 m long. The ship distribution on the waterway was combined normal (66 %) and uniform (34 %) distribution.
Figure 9: Location 1 (see Figure 8) between islands on the waterway to Sköldvik.
Macduff's Model
As it is mentioned before, to apply Macduff's model a waterway has to be approximated as a channel of which width is assumed not to change significantly when moving along the channel. No such channels exist in the Gulf of Finland. However, Macduff himself used his model for the Dover strait that is a natural channel as well; so its width is not constant either.
The modified equation of Macduff's model [Eq.( 0 )4] with C=900m, L=147m and s=20L gives the geometrical grounding probability of PG=0.9018 or 1061 annual grounding candidates, which is a quite high probability. However, if the definition of s is replaced by the new definition [Eq.( 0 )] and V=11.78 kn and a=180s, the probability would be PG = 0.0456 or 54 annual grounding candidates. The original equation of Macduff's model [Eq.( 0 )] gives the geometrical grounding probability of PG = 4.16, which is not acceptable.
As is seen, the probability in Macduff's model is very sensitive to definition of s, which requests further attention to be rational. However, since there were no registered grounding accidents for the mentioned area during the past 12 years (1996-2008) in the database of Accident Investigation Board of Finland [9] , the new definition for s seems more reasonable.
Fujii's Model
To apply Fujii's model, it needs to be considered that the channel has shoals in the middle of it. For that reason, the islands on the western side of the channel are considered to be part of the channel. Thus the new width of the channel is 1960 m (D=850 m).
According to the new definition [Eq.( 0 )], traffic density (Ï) for the mentioned location is , when the time window is set to one year. Then, according to Eq.( 0 ), the traffic volume (Ф) is . Thus, by the help of Eq.( 0 ), 0.0055 annual grounding candidates are obtained for the studied location; which would mean one grounding candidate every 182 years.
Pedersen's and Simonsen's Models
It is said that the model of Pedersen is implemented to the program called IWRAP Mk2. However for the event of checking position of the ships, it seems that the theory of Simonsen is used in the program. Nevertheless, the following calculations are made with IWRAP Mk2.
Depth curves of 10 m, 6 m, 3 m and 0 m of the area were inserted to the program. Figure 10 presents the waterway in question as it was defined for calculations in IWRAP Mk2, and Figure 11 presents the riskiest locations in the area. For the area in question, IWRAP Mk2 gave of 0.00544 grounding candidates annually when a blackout frequency of 1.75 was assumed and 0.00233 candidates if the blackout frequency of 0.75 was assumed. All grounding candidates are drift grounding candidates. Drift speed was assumed to be 1 knot. Interestingly, the first mentioned result is similar to what comes out from Fujii's improved model. However, it should be borne in mind that Fujii's presents just the annual grounding candidates, while IWRAP Mk2 presents both the annual grounding and stranding candidates together.
Figure 10. Location 1 (see Figure 9) as defined in IWRAP Mk2: grounds in black, 10 m depth curve in light gray, 6 m depth in medium gray, and 3 m depth curve in dark gray.
Figure 11. Results of grounding frequency calculation of IWRAP Mk2. The depth curves are shown white here for more clarity. A scale from light gray to dark gray is used to illustrate the grounding probability for different locations: dark gray signifies the highest number of grounding candidates in the analyzed area.
Type II Calculations
For type II calculations, an example location on the way to Sköldvik was chosen (Location 2 in Figure 8, Figure 12). Ships have to turn or they will run aground on an island about 1000 m after they have omitted to turn. On this waterway, 893 tankers navigated northwards in 2008. The largest tankers heading to Sköldvik cannot use the waterway in question and thus the length of all tankers is less than 175 m and the most common length group of tankers is 125-150 m. The average speed of the tankers approaching the island is 19.2 knots.
Figure 12: Location 2 on the waterway to Sköldvik (grounding scenario is marked with black ellipse).
Table 1 presents the number of annual grounding candidates in the studied waterway according to Pedersen's and Simonsen's models for different mean times between position checking, which results different values of ai. From Table 1, it is obvious that Pedersen's model is very sensitive to the value of ai. Simonsen has assumed the event of checking the position of the ship to be a Poisson process, and this made his model less sensitive to ai compared to Pedersen's model. Still, Simonsen has claimed that his model is also sensitive to the value of ai. However, the sensitivity of Simonsen's model will decrease when the larger values are used for ai; which was predictable as he uses Poisson process. As a conclusion, it should be noted that hardly even professional navigators can estimate ai so exactly that it would not give a large uncertainty to the results. (Ylitalo et al. 2008)
Table 1. Number of type II grounding candidates with different time interval of checking the position.
Mean time between position checking
30 s
60 s
90 s
120 s
150 s
180 s
210 s
Annual grounding candidates according to Pedersen's model
0.02
37.68
502.38
1834.37
3989.88
6697.94
9697.12
Annual grounding candidates according to Simonsen's model
30.57
165.21
289.94
384.10
454.7
508.84
551.41
Conclusion and Further Research
Calculations of type I have showed that four used models give different results for the number of annual grounding candidates. Since, for the studied area, there are no available databases about the near-missed cases and no registered actual groundings in the database of Accident Investigation Board of Finland for years 1996-2008, it is hard to say which model gives more accurate results than the others. In general, it seems that the chosen location is not a high risk area for grounding accident. Thus low probability for grounding candidates seems more sensible.
Calculations of type II show Pedersen's approach gives more grounding candidates than the actual traffic (893 tankers) for mean times between position-checking longer than 100 s. Although it is hard to define an exact value for mean time between position-checking, using larger values than 100 s is not rare; for instance IWRAP Mk2 uses 180 s as its default value. In addition, Simonsen's model is less sensitive to the values of ai. Thus Simonsen's approach is more rational than Pedersen's.
The main issue in analyzing the risk of grounding is that by today's knowledge, a holistic and precise model which could describe the reality, if not say impossible, is at least a really hard goal to achieve. Although Macduff and Fujii were the pioneers in the geometrical analysis of the grounding probability, it is shown in this paper that their methods have some weaknesses. So far, it can be said that Simonsen's model, which is based on Pedersen's first idea, is a more rational and completed geometrical model than other mentioned models. Simonsen considers (like Pedersen) not only the class, dimensions and velocity of the vessels, but also the distance between position-checking, which is more or less a human factor in navigation.
One another important issue is that geometrical probability and causation probability are closely linked to each other when calculating the grounding probability. When estimating causation probability, the used geometrical model has to be taken into consideration. The simplest way to calculate the causation probability is to divide the number of actual groundings by the number of grounding candidates, like what (Macduff 1974) and (Fujii et al. 1974) did. Thus, the causation probability includes a close link to the used geometrical model. Therefore, the causation probability cannot be directly used with other geometrical models. Nor can it be compared with other causation probabilities without paying attention to the geometrical models they are made to be used with.
It seems there is no place for more improvement in Macduff's and Fujii's models, based on their first theories. Thus, in order to find a better model for estimating the ship grounding probability, either Simonsen's model should be improved or another method, for instance using fuzzy ship domain, should be introduced. Introducing a 3D fuzzy ship domain, with proper shape and size, for ship grounding probability analysis is the idea that authors consider as their next step in their researches.
