Probabilistic Analysis of Basic Causes of Vessel–Platform Allision Accidents

Abstract

This paper proposes a methodology to estimate the probability of basic causes of allision accidents between vessels and offshore platforms that overcomes the problem of data scarcity required for causal analysis. The approach uses information derived from incidental data and expert elicitation, processed by a multiple attribute utility method and hierarchical Bayesian analysis. First, the methodology is detailed, briefly describing the adopted approaches. A dataset of allision incidents provided mainly by the UK Health and Safety Executive and other agencies is prepared. The features of the incidents’ causation in terms of the causal factors and basic causes are presented and discussed. A novel scheme is proposed to evaluate the annual occurrence rates of basic causes of accidents from the relative importance of each basic cause derived by the Deck of Cards method. Then, a hierarchical Bayesian analysis is conducted to predict the posterior distribution of the occurrence rate of each basic cause in the time frame under analysis. The proposed holistic methodology provides transparent estimates of allision causation probabilities from limited and heterogeneous datasets.

1. Introduction

The offshore and maritime industries have witnessed several significant accidents and disasters throughout history, such as fires and explosions, hazardous material releases, and crane accidents, responsible for several fatalities, structural damage to facilities and environmental damages [1,2,3,4]. Major accidents such as the Pipe Alpha, the BP Deepwater Horizon and the Cidade de São Mateus FPSO have encouraged hazard-prone industries to apply risk assessment techniques for preventing accidents and mitigating their consequences [5,6,7]. Thus, an adequate prediction of the likelihood of potential accidents and their causes is highly significant for improving the overall safety of offshore facilities [8,9]. The probability of such accidents can be predicted using a proper methodology and evidential data concerning their causes [10,11]. The adopted methodology should comprehensively cover the issues of data sparseness and the relationships between causal factors and accidents [12].

Offshore accident scenarios have different characteristics, risks and consequences [13,14,15,16]. The greatest attention is paid to understanding and modelling fire and explosion risks in offshore facilities [5,17,18,19]. However, other accidents, such as allisions, may pose other risks with potentially high consequences [20].

The oil and gas and other hazardous industries are encouraged to report major accidents as well as precursors to the respective authorities [21,22]. For example, the Norwegian Ocean Industry Authority (Havtil), formerly known as the Petroleum Safety Authority (PSA) of Norway [1], the Health and Safety Executive (HSE) [2] of the UK, and the Bureau of Safety and Environmental Enforcement (BSEE) [22] of the US are some major onshore and offshore agencies responsible for controlling the risk around their legislative area and adopting safety measures.

Allision is an accident scenario of a moving ship hitting a stationary object, such as an offshore oil and gas installation, which may cause severe damage to the installation, human fatalities and environmental damage [7,23]. The term allision differs from collision as the latter is used when considering an impact between two moving vessels, which is not in the scope of the present study. Ship collision is one of the most studied maritime accidents [24,25], subsequently covered by several frequency estimation models using statistics and expert input [26]. On the other hand, less attention is paid to allision due to its relatively rare occurrence. Ship accident databases [3,4,27] generally code allision accidents under the category of “contact (with fixed and floating structure)” and “collision”; however, the choice of the taxonomy is a matter of preference of the authors.

Offshore regulators such as Havtil [1] and the Health and Safety Executive (HSE) [20] are highly concerned about the allision risk as it might be conservatively assessed, and more knowledge is needed to determine the actual allision risk. Both the authorities categorize allision as a major accident hazard with dramatic consequences. Allision risk assessment has been evolving for more than three decades [28]. The research community and industries have taken the initiative to understand, model, prevent and mitigate allision scenarios. For example, to avoid allision incidents, guidelines for contingency plans and measures are provided by the UK Oil and Gas Industry Association [7]. Furthermore, the guidelines state the importance of some collision detection systems.

Some researchers and industrialists have also examined this phenomenon with different aspects and techniques. In an early study, Zhao et al. [23] identified six causes of allision: unawareness of the existing platform, position fixing error, watch-keeping failure, radar failure, equipment failure and human error. However, their probabilistic collision model is focused on human error, while the causation is based on fuzzy mathematics. An allision model called the Ship Offshore Platform Collision Risk Assessment (SOCRA) has been developed, incorporating only two scenarios: ramming and drifting collisions [20]. The COLLIDE model contemplates inaccuracies in navigation due to equipment failure, weather, and human error [29]. More commercial software, like CRASH by DNV, COLRISK by Anatec, and MANS from MSCN (Netherlands), exist that use shipping traffic databases to characterize the probability of allision [7,30]. These tools are prone to model assumptions and should be subjected to substantial improvements as technologies advance [30,31,32].

Ship traffic databases, particularly data from the Automatic Identification System (AIS), have been extensively and efficiently used for probabilistic allision assessment [31,32]. Recently, Hörteborn and Ringsberg [33] studied the ship bridge allision probability by identifying accident scenarios using ship manoeuvring and AIS data. However, such simulation-based models only focus on a specific extreme event, leaving many other important risk factors partially or totally uncovered. Nevertheless, some other probabilistic allision models have been developed, deliberating only specific aspects or scenarios. For example, Chen and Moan focused on FPSO and tanker allision during offloading operations using drive-off frequencies [34].

Several studies have modelled allision scenarios based on the likelihood of a vessel on a collision course and the probability that the vessel does not take collision avoidance actions, the so-called causation probabilities [34,35,36]. The majority of allision risk models consider the spatial distribution of ship traffic and the offshore installation position [28]. These models have inherent advantages, such as the phenomenon’s geometric representations and dynamic characteristics using current traffic data. The limitation lies in the superficial estimation of the causation factors and causation probabilities, such as in the studies [33,37].

In a qualitative assessment, Kvitrud [38] emphasized the safety culture on the vessel, inadequate personnel training, complex equipment and watch-keeping failures, as the foremost causes of allision. Despite the advances in technology, such as ship automation, the allision incidents have not been reduced accordingly, as watch-keeping failures (human errors) are the main issue associated with allision [39]. Oltedal [40] directed his study beyond human errors and identified the underlying factors influencing allision from the analysis of six PSA reports.

The causal relationship among leading events and their precursors is often portrayed via probabilistic techniques like fault trees, bow ties, and event trees, which have been more recently replaced by more flexible modelling tools such as Bayesian belief networks. In the context of this study, a pioneering study by Hassel et al. [31] proposed a Bayesian network-based risk model that incorporates risk-influencing factors causing allisions. However, the quantification of all the nodes has been carried out by an expert panel, making the model highly subjective to the opinions and knowledge of the experts. It is argued that the Delphi method is highly useful when experts’ opinion is the only available source of information [41,42]. Critics claim that this process is too time-consuming and contains uncertainties due to experts’ elicitation [43].

Recent studies have recognized that allision risk assessment practices are exceedingly conservative. On the one hand, the traffic patterns before and after the platform installation may pose a significant threat to offshore platforms [31,32]. On the other hand, a successful probabilistic allision assessment depends on the identification of substantial causal elements [30]. In a comprehensive review, Xiao et al. [28] have revealed that most commonly used allision models often miss out on important risk factors for the holistic depiction of this scenario. They further emphasized that causal factors, particularly human factors and the influence of weather conditions, must be analyzed in depth to predict the probability of incidents.

Allision is difficult to avoid as the procedural violations drift into normal operational behaviour and, therefore, are difficult to include in organizations’ safety management [40]. Moreover, analytically developed models using AIS focus on the final event probability of allision, emphasizing fewer causal factors, while BN-based risk models are associated with high uncertainty in the nodes’ prior and conditional probabilities.

An accident can evolve from its precursors, like initiating events or basic causes. In risk analysis, precursor-based studies are conducted within two scopes: (i) to develop major accident indicators based on the accident precursors and (ii) to develop likelihood functions that are further utilized in Bayesian analysis. This study focuses on the former application. If enough causal data with high fidelity are available, the probability of basic causes can be calculated using statistical approaches such as the maximum likelihood estimator. A dedicated allision analysis of many thousands of accumulated offshore installation years is provided by the HSE [44,45]. Often, these reports provide information on the causal factors (immediate causes) and some basic causes of the accidents but lack information regarding the root causes [46]. However, the historical databases [1,4,27] usually suffer from a degree of under-reporting, inconsistent taxonomy and inaccuracy of information [24].

Expert elicitation can provide reasonable and high-fidelity results in cases where causal data are uncertain and limited [47]. Besides Bayesian theory and fuzzy logic, some available methods in the literature can be applied to include expert opinions on risk assessment and decision-making, such as cloud modelling [48], analytical hierarchy process [49], paired comparison [50], maximum expert consensus model [51], TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) [52], Dempster–Shafer (D–S) evidence theory [53], and SHELF—Sheffield elicitation framework [54]. These methods involve allocating weights to quantify the relative significance and ranking to create an order of preference for indicators. The stated methods are very demanding from the cognitive point of view. Some of these methods (e.g., AHP) require a prefixed evaluation scale with a well-defined number of levels. These issues make them relatively abstract and intricate for an expert’s judgment.

The Deck of Cards method (DCM) is one simple way that has been used to define the criteria weights of outranking Multicriteria Decision Aid methods [55,56]. Corrente et al. [55] have conferred some salient features of the DCM and pointed out its advantages over conventional multiple-criteria judgement procedures. The DCM can designate a pairwise comparison of indicators concerning given criteria regarding the number of blank cards expressing differences in attractiveness. The DCM represents a direct rating (via blank cards) of the difference in attractiveness between consecutive indicators, making it easy to apply and translate judgments into numerical values. Visual interpretation allows experts to express judgments more freely and precisely than in the relatively abstract approaches of other methods. Moreover, the method can use background information (which can be traced from observed data) to rank indicators, making it less hypothetical and therefore more aligned with reality. Consequently, the method allocates expert elicitation systematically using a simple approach and its predictions do not deviate significantly from the observations.

Quantitative allision risk models may require comprehensive and quality data for platform and vessel safety [57]. The present work aims to develop an incidental and expert-based methodology for the probability estimation of basic causes of allision accidents. In the paper, elicitation of expert judgments is used to calculate prior probabilities derived from the Deck of Cards method coupled with observed data. The hierarchical Bayesian analysis (HBA), which has been widely adopted in probabilistic risk analysis where data uncertainty and source variability are major issues [58,59], is then employed to derive the posterior predictive distributions of the occurrence frequency of the basic events. A detailed review of HBA to address the population variability in the risk assessment community is provided by Kelly and Smith [60].

The rest of this paper is organized as follows. Section 2 presents the methodology developed in this paper. An overall discussion of the results and the implications of the findings by the application of the methodology are presented in Section 3. Finally, the conclusions and the important findings of this study are provided in Section 4.

2. Methodology

This section describes the developed methodology for identifying the causation probability of allision of a vessel and offshore installation. This methodology has two major parts: (1) ranking of the allisions’ basic causes from incident data collection and (2) posterior probability estimation of the basic causes of allision accidents. Figure 1 shows the methodology outline with the two main parts in different colours, and the procedure is explained as follows.

First, raw data are collected from offshore databases and published statistical investigation reports on allisions. The analysis of data can reveal significant insights about the circumstances of allision. In particular, from the analysis of data, the immediate causes (causal factors) and their basic causes are identified. Based on the number of occurrences of each basic cause, they can be ranked from least to most observed. From understanding the causation of an accident, more basic causes are identified by experts’ elicitation. For this purpose, a well-known decision-making tool called the Deck of Cards method (DCM) [55,56] is used to assign normalized weights to these basic causes. A novel scheme is proposed to transform accident occurrence into the occurrence rates of basic causes by applying the relative importance of each basic cause deduced from the Deck of Cards method. From the collected data, the annual frequency of accidental events (allision) is gathered, and when multiplied by the weights, the occurrence rate of each basic cause for each year is deduced. The probability of basic causes can be estimated as the posterior distribution mean of the annual occurrence rates using hierarchical Bayesian analysis. This scheme and the methods involved are detailed in the following subsections.

2.1. Deck of Cards Method

The DCM has been used to define the criteria weights of outranking Multicriteria Decision Aid (MCDA) methods. The DCM allows the construction of interval and ratio scales, which can be used to determine the weights of criteria [56] and to model the strength of preference between different levels of criteria scales [55]. The DCM ranks all the considered elements from the least important to the most important. Then, the strength of preference is established by adding blank cards between consecutive levels and fixing the ratio between the weight of the most important element and that of the least important one. In this paper, the DCM is adopted to estimate the normalized number of occurrences of each basic cause based on a ranking derived from objective data and, therefore, is not based on unrealistic assumptions. This way, the method uses background information and allocates expert elicitation systematically; thus, its predictions do not deviate significantly from the observations. A brief overview of this method is presented here; however, for more details readers may refer to [55,56].

2.2. Hierarchical Bayesian Analysis (HBA)

Bayesian analysis is based on Bayes’ theorem, which defines the probability of an event based on prior information related to the event and observed data. Mathematically, a traditional Bayesian model is specified in the present context as

$$p\left(\theta |\lambda \right)\propto p\left(\lambda |\theta \right)p\left(\theta \right)$$

(1)

where $\lambda$ is observed data, θ is the parameter of the model that can produce the data, $p\left(|\right)$ is the probability density function of given conditions, $p\left(\theta \right)$ is the prior probability of θ, $p\left(\lambda |\theta \right)$ is the likelihood function and $p\left(\theta |\lambda \right)$ is the posterior distribution of θ.

The hierarchical Bayesian analysis (HBA) helps in understanding and solving problems involving multi-parameters at multiple levels. In a hierarchical Bayesian model, the observed outcomes ${\lambda }_i$ of a given dataset i (i = 1…n) are modeled conditionally on certain parameters θ_t, which themselves depend probabilistically on higher-level parameters, called hyperparameters or population parameters $(\alpha ,\beta$).

The observed dataset vector ${\lambda }_i$ can be used to estimate aspects of the population distribution of the required parameter vector θ_i even though these are not themselves observed. In a hierarchical model, ${\lambda }_i$ are modelled conditionally on θ_i, which are probability specified using a population distribution governed by parameter vectors α and β. Now, the parameter θ_i is assumed to be symmetrical or similar to the prior distribution across the data. Therefore, the parameters (θ_i, i = 1…n) are considered exchangeable in their joint probability distributions. This is mathematically simplified as

$$p\left(\theta |{\alpha }^{\prime },{\beta }^{\prime } \right)=\prod _{\mathrm{i}=1}^{n}p\left({\theta }_i|{\alpha }^{\prime },{\beta }^{\prime }\right)$$

(2)

Since α and β are unknown, the distribution of θ must average over uncertainty in α and β:

$$p\left(\theta |\alpha ,\beta \right)=\int \left({\prod }_{\mathrm{i}=1}^{n}p\left({\theta }_i|\alpha ,\beta \right)\right)p(\alpha ,\beta )d\alpha d\beta$$

(3)

The above form can capture exchangeability via the mixture of independent identical distributions. Since α and β are unknown, they can be given their prior distribution $p(\alpha ,\beta )$. The Bayesian posterior distribution is of the vector ($\alpha ,\beta$, θ), with a joint prior distribution as

$$p\left(\alpha ,\beta ,\theta \right)=p\left(\theta |\alpha ,\beta \right)p(\alpha ,\beta )$$

(4)

and the joint posterior distribution of α and β is

$$p\left(\alpha ,\beta ,\theta |\lambda \right)\propto p\left(\lambda |\theta ,\alpha ,\beta \right)p\left(\theta |\alpha ,\beta \right)p(\alpha ,\beta )$$

(5)

where $p\left(\lambda |\theta ,\alpha ,\beta \right)p\left(\theta |\alpha ,\beta \right)$ is the likelihood function. It is worth noting that the diffuse hyper priors $\alpha ,\beta$ are independent before data observation and affect $\lambda$ through θ.

The solution of the posterior distribution presented in Equation (5) involves a multi-dimension integral with as many dimensions as $\alpha ,\beta$. Furthermore, the multi-stage prior distributions used in the hierarchical Bayesian model are also difficult to calculate. Therefore, approximation techniques such as the Markov chain Monte Carlo (MCMC) may be best suited for providing high-fidelity solutions. In MCMC simulations, θ values can be drawn from approximate distributions and then can be adjusted to better approximate the target posterior distribution.

3. Results and Discussion

3.1. Data Collection

This study uses two large and descriptive sets of UK Continental Shelf (UKCS) allision incident data reports [43,44] as primary information to understand and characterize the allision accidents. The HSE is a pioneering organization based in the UK responsible for health and safety-related issues. The main objective is to obtain complete statistics for incidents on offshore platforms engaged in oil and gas activities on the UKCS from 1975 to 2016.

More information from the Havtil [1] and BSEE [22] databases is gathered to understand the trend of reporting for allision incidents in the offshore sector. Furthermore, broad ship accident data, available at [3,4,27], have also been studied. These reports (available in the public domain) reveal basic accident indicators such as the number of occurrences, ship losses, and casualties with respect to the time and ship type, which are results of superficial scrutiny, yet in-depth causal factors (immediate causes) and basic causes of the accidents are rarely described. Moreover, the detailed literature review of scientific papers conducted and discussed in the Introduction section provides credible allision accident scenarios.

3.1.1. Causal Factors

This section aims to understand the basic information about the key circumstances leading to allision accidents. Allision incidents are commonly grouped into powered and drifting collisions; however, the present study does not discriminate incidents by groups and aims to identify overall causal factors [7]. This is to create a wider view of the allision scenario rather than showing the chain of events of a particular case. Data from all the reported incidents have been analyzed, including those categorized as severe, moderate, or minor, and for all the vessel and installation types. For this analysis, a population of 557 incidents is gathered. The initial process identified the immediate causes (causal factors) of allision accidents. The results of the analysis are shown in Figure 2 as a distribution of the causal factors of allision incidents. The main contributors to accidents are human errors (30%), followed by equipment failures (22%). Finally, 15% of allision incidents involved the role of external factors. This study attempts to identify the causal factors behind such incidents; however, due to the limited description of incidents in the source data, 33% of the immediate causes remain unknown. This fact justifies the proposed methodology as there is huge uncertainty in the causal analysis of data, and it may not be used with high fidelity in decision-making. Another important point is that the data are limited and only represent a small proportion of worldwide accidents.

3.1.2. Identification of Basic Causes

A good understanding of the basic causes leading to an accident is essential in any risk assessment. The database is analyzed thoroughly to obtain generic allision causal scenarios based on balanced hazard identification and analysis of historical incidents. The basic cause is the last identified factor in the investigation reports and may not be a root cause that is the most fundamental set of circumstances (may be active or latent) that lead to an event. From the recorded 557 incidents, basic causes are identified only for 375 incidents (Figure 2). The reported basic causes have been slightly modified as per the authors’ opinion and understanding of the scenario.

Figure 3 presents the distribution of basic causes for equipment failure. Engine failures are mostly due to control equipment failures, constituting 21%, while engine power failures include 10% of the overall equipment failures. Failure of Dynamic Position (DP) equipment involves control failure (6%), computer failure (1.6%), electrical failure (1.6%), and remote control failure (0.8%). Some other important pieces of equipment that need to be paid high attention to as prone to failure are thruster, steering, mooring, electrical and power-related equipment. Other causes (propeller, crane, rudder, etc.) are distributed around 0.8%.

Figure 4 presents the basic causes of external factors (a) and human errors (b), with each figure split into two frames that are distributed based on observed data (Data) and expert elicitation (Expert). As seen in Figure 4a, external factors are in the majority of events caused by adverse weather conditions (82%), followed by anchor dragging (16%) and obscured vision (2%). It is deemed important to break down further possible weather elements that are important for allision scenarios. As discussed above, secondary databases are consulted, which identify that heavy rain, thick fog, extreme waves and wind conditions are the main weather conditions causing allision. The quantitative details are derived from expert elicitation, as discussed and implemented in the next section.

Figure 4b depicts the common human errors present in allision incidents. Three-fourths of the errors fall under misjudgments, while the other important ones are operator error (10.8%) and watch-keeping failure (9%). Similar to the above scheme, misjudgments are further classified as lack of awareness, lack of knowledge, miscalculation and improper communication. The present scheme could be further applied to find the precise nature of errors or root causes as a second-layer investigation; however, this would increase the complexity of subsequent calculations without offering any methodological benefits. Thus, it is not deemed necessary to go further in this study and the additional step will be the scope of future developments.

3.2. Application of the DCM

This section is devoted to the estimation of the normalized weights that stand for the relative importance of each basic cause of allision incidents. One approach can be a simple statistical distribution, as seen in Figure 3. However, the DCM can be implemented to include more apparent causes (Figure 4—expert frame) and expert opinion in quantifying the relative importance of causes.

There are 42 identified basic causes (under three causal factor groups) to be judged using the DCM. First, they are ranked in consecutive levels l₁ to l₄₂ (where l_h corresponds to basic cause h = 1 to 42) based on the number of occurrences. Clearly, l₁ is the least apparent basic cause, while l₄₂ is the most prominent basic cause. Based on the expert elicitation, some blank cards are included between consecutive levels (basic cause). The number of blank cards sets up the relative importance of a basic cause over another basic cause. Next, a comparison table is constituted with the basic cause and number of inserted cards (in bold), as shown in

Table 1. Comparison tables of the DCM.

(a) Table with inserted cards
	l1	l2	l3	l4	.		l42
l1		1
l2			2
l3				2
.					.
.						.
l41							6
l42
(b) Table filled using the consistency condition
	l1	l2	l3	l4	.	.	l42
l1		1	4	7			102
l2			2	5			100
l3				2			97
.					.
.						.
l41							6
l42

a), where “1” is the number of blank cards placed between l₁ and l₂, while “6” is the number of blank cards placed between l₄₁ and l₄₂.

Then, the remaining elements (e_h,k h—row number, k—column number) of the comparison table are filled using the consistency condition [55] given as e_h,k = number of blank cards allocated in row h + e_h+1,k + 1. For example, e_1,3 = 1 + 2 + 1 = 4. Consequently, all the blank entries can be filled in the comparison table as shown Table 1b).

Due to space limitations, only some elements of the table are shown. In the next step, an expert must specify the relative importance as how many times ($z$) the weight of the most important basic cause (l₄₂) is higher than the least important basic cause (l₁). In the present case, it can be presented as $z=u\left(l_{42}\right)/u\left(l_1\right)$, where $u\left(l_h\right)$ is the utility value of a cause $l_h$.

The next step is evaluating the value of the unit ($\alpha$) among levels as follows:

$$\alpha =\frac{u\left(l_{42}\right)-u\left(l_1\right)}{m}$$

(6)

where m is the number of units between the lowest level (l₁) and the highest level (l₄₂). In the present case, m = e_1,5 + 1 = 102 + 1 = 103. Let the utility values of the two extreme levels be given as $u\left(l_{42}\right)$ = 4.2 and $u\left(l_1\right)$ = 1. Consequently, $z=$ 4.2 and $\alpha$ = 0.031.

Finally, the utility values for other levels can be calculated as follows:

$$u\left(l_2\right)=u\left(l_1\right)+\left(e_{12}+1\right)\times \alpha =1+2\times 0.031=1.062$$

$$u\left(l_3\right)=u\left(l_1\right)+\left(e_{13}+1\right)\times \alpha =1+5\times 0.031=1.155$$

$$u\left(l_{42}\right)=u\left(l_1\right)+\left(e_{14}+1\right)\times \alpha =1+103\times 0.031=4.2$$

Using the calculation of the utility values, the normalized values $w\left(l_h\right)$ at each level (=weights of basic causes) can be determined as $w\left(l_h\right)=u\left(l_h\right)/\sum u\left(l_h\right)$. For example, in the present case,

$$w\left(l_1\right)=u\left(l_1\right)/\left\{u\left(l_1\right)+u\left(l_2\right)+u\left(l_3\right)+u\left(l_4\right)+\dots +u\left(l_{42}\right)\right\}=1/102.8=0.0097.$$

Similarly, $w\left(l_2\right)$, $w\left(l_3\right)$ … $w\left(l_{42}\right)$ are estimated as 0.0103, 0.0112, … 0.0409, respectively. Appendix A details the important results in terms of the Ranking $\left(l_h\right)$, Utility values $u\left(l_h\right)$, and Normalized values $w\left(l_3\right)$ from the analyses presented in this section. This study restricts the number of basic causes to the ones found in the database. However, in a further theoretical approach, more apparent basic causes will be included. For example, any additional (non-evidential) cause can be easily introduced at any level based on expert opinion. This convention can be treated as an inherent advantage of the present methodology, where simple ways can readjust the analysis and subsequent results.

3.3. Estimation of the Occurrence Rate of Basic Causes

From the scrutiny of the allision accident database, the annual incident (allision) occurrence can be calculated. Since the number of platforms is not uniform across all the operational years, the number of incidents is normalized with the number of platforms in each year. Let N^t be the annual occurrence rate of allision as the ratio of the number of times allision incidents occurred (A^t) to the number of installations operational (I^t) in a year t (t = 1 to 16 years, as in

Table 2. Number of allision incidents and installations from 2000 to 2015, according to [44].

t	Year	Number of Allision Incidents (At)	Number of Installations (It)	Annual Occurrence Rate (Nt)
1	2000	18	300	0.060
2	2001	12	307	0.039
3	2002	10	308	0.032
4	2003	6	311	0.019
5	2004	4	313	0.013
6	2005	7	314	0.022
7	2006	8	315	0.025
8	2007	12	331	0.036
9	2008	8	337	0.024
10	2009	4	338	0.012
11	2010	5	332	0.015
12	2011	7	332	0.021
13	2012	4	335	0.012
14	2013	6	337	0.018
15	2014	4	340	0.012
16	2015	3	331	0.009

). It is worth noting that an accident can be considered as its precursor. Following this rationale, it is proposed that the annual occurrence rate of allision (N^t) is distributed to all the basic causes (h = 1 to 42 basic causes) in terms of the annual contributing occurrence rates (${\lambda }_h^t$, h = 1 to 42 basic causes, t = 1 to 16 years) such as:

$${\lambda }_1^t+{\lambda }_2^t+{\lambda }_3^t+\dots {\lambda }_{42}^t=N^t$$

(7)

The above equation corresponds to time “t”, which can be normalized as follows:

$$\frac{{\lambda }_1^t}{N^t}+\frac{{\lambda }_2^t}{N^t}+\frac{{\lambda }_3^t}{N^t}+\dots \frac{{\lambda }_{42}^t}{N^t}=1$$

(8)

The ratio of the left-hand terms is now independent of the time and synonymous with the normalized weight of each basic cause on the overall causation of the final event. There can be multiple ways of deducing these weights, such as the results of simple statistical analysis. However, this study adopts an expert-based approach using the DCM, as explained in the previous section, which implies the following:

$$w\left(l_1\right)+w\left(l_2\right)+w\left(l_3\right)+\dots w\left(l_{42}\right)=\frac{{\lambda }_1^t}{N^t}+\frac{{\lambda }_2^t}{N^t}+\frac{{\lambda }_3^t}{N^t}+\dots \frac{{\lambda }_{42}^t}{N^t}=1$$

(9)

As the weights are already obtained in the previous section, the annual contributing occurrence rates can be calculated for each cause per year as ${\lambda }_h^t=w\left(l_h\right)\times N^t$. For the sake of demonstration and simplicity, a smaller set of annual allision frequency data is obtained from the HSE [44], as shown in Table 2. It is to be noted that the last 16-year evidential data are used here to keep the calculations simple. Another important piece of information perceived from Table 2 is the decline in the number of allision incidents in recent years. This is mainly due to technological advances in navigation and traffic monitoring, including improvements in navigation systems and communication technologies, the development and implementation of better risk management practices, and improved training and education for maritime and offshore operation personnel. However, these aspects are not in the scope of the present study.

Given the data gathered and analyzed with the adoption of the above scheme, the annual contributing occurrence rates for all the basic causes are estimated and a portion of the overall table is illustrated in

Table 3. Annual contributing occurrence rates for all the basic causes from 2000 to 2015.

t	Year	${\mathit{\lambda }}_1^{\mathit{t}}$	${\mathit{\lambda }}_2^{\mathit{t}}$	${\mathit{\lambda }}_3^{\mathit{t}}$	…	${\mathit{\lambda }}_{42}^{\mathit{t}}$
1	2000	5.84 × 10−4	6.20 × 10−4	6.74 × 10−4		2.45 × 10−3
2	2001	3.80 × 10−4	4.04 × 10−4	4.39 × 10−4		1.60 × 10−3
3	2002	3.16 × 10−4	3.35 × 10−4	3.65 × 10−4		1.33 × 10−3
4	2003	1.88 × 10−4	1.99 × 10−4	2.17 × 10−4		7.88 × 10−4
5	2004	1.24 × 10−4	1.32 × 10−4	1.44 × 10−4		5.22 × 10−4
6	2005	2.17 × 10−4	2.30 × 10−4	2.51 × 10−4		9.11 × 10−4
7	2006	2.47 × 10−4	2.62 × 10−4	2.85 × 10−4		1.04 × 10−3
8	2007	3.53 × 10−4	3.75 × 10−4	4.07 × 10−4		1.48 × 10−3
9	2008	2.31 × 10−4	2.45 × 10−4	2.67 × 10−4		9.70 × 10−4
10	2009	1.15 × 10−4	1.22 × 10−4	1.33 × 10−4		4.84 × 10−4
11	2010	1.47 × 10−4	1.56 × 10−4	1.69 × 10−4		6.15 × 10−4
12	2011	2.05 × 10−4	2.18 × 10−4	2.37 × 10−4		8.61 × 10−4
13	2012	1.16 × 10−4	1.23 × 10−4	1.34 × 10−4		4.88 × 10−4
14	2013	1.73 × 10−4	1.84 × 10−4	2.00 × 10−4		7.27 × 10−4
15	2014	1.14 × 10−4	1.22 × 10−4	1.32 × 10−4		4.81 × 10−4
16	2015	8.82 × 10−5	9.36 × 10−5	1.02 × 10−4		3.70 × 10−4

. The elements in Table 3 present the expected occurrence of each basic cause per installation year from 2000 to 2015.

3.4. Estimation of the Probability of Occurrence Using HBA

The occurrence frequencies of adverse events in a given period (number of ship collisions per year) are often used as probabilities in the maritime industry [61]. This approach is supported by the International Maritime Organization (IMO) methodology known as the Formal Safety Assessments’ definition of risk [62]. This study uses this convention that transforms the annual contributing occurrence rates of basic causes into the occurrence probability using their mean posterior distribution derived from HBA.

Lunn et al. [63] have utilized probabilistic graphs to describe the complex statistical dependence among related parameters. These are directed acyclic graphs in which nodes may represent data/observations, variables and constants, while directed edges represent their relationship.

The annual data about the contributing occurrence rates (${\lambda }_h^t$) of basic causes (h = 1 to 42) can be grouped into t = 1 to 16 (for the years 2000 to 2015) data groups corresponding to each year. Figure 5 shows an HB modelling scheme used in the present context. It should be noted that the HBA is conducted for each basic cause “h” (=1 to 42) independently, and therefore, the subscript “h” is deliberately omitted from the following discussion.

As shown in Figure 5, the hierarchical Bayesian model uses groups of exchangeable information to deduce θ^t, which is similar but does not need to be identical. In the present study, θ^t represents the occurrence probability of a basic cause for the year “t”. These θ^ts are exchangeable and are assumed to be a sample from a common population distribution; therefore, they are assigned common prior α and β, known as the population level parameter. This approach anticipates that the samples in datasets often form clusters or groups within which some properties are shared. Thus, through the structure (as illustrated by the dashed lines in Figure 5), information is borrowed from one group to another while estimating a group-specific parameter. A vital feature of this approach is that the observed ${\lambda }^t$ data within a group can be used to estimate the population distribution (α and β) of θ^ts even though the values of θ^ts are not themselves observed. Further details regarding the applications of HBA can be derived from references [63,64,65]. The HBA generates the most suitable values for the parameters of interest (e.g., θ^t) involving minimum uncertainty [63,66,67,68].

The probability of a basic cause (θ^t) can be derived from the contributing occurrence rates (${\lambda }_h^t$, h = 1 to 42 basic causes, t = 1 to 16 years) observed against the time using the HBA.

The θ^t is considered to follow a gamma distribution with hyperparameters α and β, that is, θ^t ~ gamma(α, β). Furthermore, the hyperparameters are also assumed to follow diffusive gamma distributions as α ~ gamma(0.0001, 0.0001) and β ~ gamma (0.0001, 0.0001).

Assuming the basic causes to be an event-based process, the contributing occurrence rates in each time interval (year) follow a Poisson distribution as ${\lambda }^t$ ~ Poisson (θ^t, t). The corresponding values of ${\lambda }^t$ for each basic cause are listed in Table 3. The HBA doodle is depicted in Figure 6, in which all the nodes are stochastic, besides mu [t] and T [t], which are logical and constant nodes. The corresponding OpenBugs [63] script is as follows:

model
	{
	for(t in 1:16 ) {
	mu [t] < −theta [t] * T [t]
	theta [t] ~ dgamma (alpha, beta)
	lambda [t] ~ dpois (mu [t])
		}
	alpha ~ dgamma (0.0001, 0.0001)
	beta ~ dgamma (0.0001, 0.0001)
	theta.avg ~ dgamma(alpha, beta)
	}

Noting that prior distributions are assigned for α and β, the model samples from their joint posterior distribution. This is because the base node “lambda [t]” and diffused parameters’ nodes “alpha” and “beta” are in a convergent direction (see Figure 6). Therefore, when data are detected via initialization of the node “lambda [t]”, the nodes “alpha” and “beta” become conditionally dependent on their joint posterior distribution. Subsequently to the above realizations, the posterior distribution of “theta” can be deduced as $\overline{\theta }$ ~ gamma (α, β), which is the mean of all the posteriors of θ^ts. The present methodology assumes that this posterior distribution can be used as the probabilities of basic causes. Figure 7 illustrates the posterior distribution $\overline{\theta }$ inferred as the probability density function of basic causes—Untangling nets (i.e., h = 2). The complete results are presented in the last column of Appendix A.

The present analysis is based on data retrieved from a UK-based database, while the methodology can be adjusted for other regions. For example, instead of data mining in Section 3.1, the analyst can directly advance to Section 3.2. A basic cause more prominent in a region can be put forward by changing the ranking, or an additional cause can be inserted. The rest of the steps could be easily followed, and the corresponding results can be obtained. Due to data limitations, it is not viable to break down the causes into specific datasets, such as according to the installation type, vessel type, damage severity and geographical location. Still, this methodology imparts confidence with which probabilistic assessment can be commenced to a level representing the best-case scenario so far as the quality of the data.

4. Conclusions

This paper presents a methodology that can systematically estimate the probabilities of basic causes of allision accidents based on limited and non-homogeneous data sources. The methodology consists of two stages, starting from the collection of incident data to the basic causes’ probability estimation using the hierarchical Bayesian analysis method. A database is developed from the HSE that contains a compilation of vessel-platform collisions known as allision incidents on the UK continental shelf. Since such data suffer from under-reporting, a scheme of data enrichment is suggested via expert elicitation using the Deck of Cards method. The objective of the proposed approach is to obtain the relative importance (or normalized weights) of basic causes of the overall allision causation.

Later, the occurrence of allision accidents is understood in terms of accident precursors, and this is dealt with by transforming the occurrence frequency of allision into the contributing occurrence rates of its basic causes. Allision occurrence data from 2000 to 2015 are adopted and the corresponding annual occurrence rates of the basic causes are calculated using the DCM-generated normalized weights. Finally, hierarchical Bayesian analysis is employed on the annual groups of data to derive the posterior distribution of the occurrence rates of the basic causes. The mean values of the posterior distribution are regarded as the probability of occurrence.

The results of this study emphasize the importance of data, expert input and formal probabilistic approaches in the development of probabilistic allision scenarios. The current study is limited to identifying basic causes, which are the last identified causes in the databases. However, there is scope to probe further into the root causes, which can also be quantified for their occurrence probability using this methodology. This work has demonstrated an efficient scheme for probabilistic analysis of allision incidents; however, it can very well be implemented for other incidents with low occurrence rates. Moreover, this methodology is an efficient probabilistic approach that can be applied to other heterogeneous data sources, such as to different databases and geographical locations, among others.