1. Introduction

The occurrence of accidents in complex systems, such as offshore and onshore oil and gas processing plants, power plants, and chemical process industries, is financially expensive because the accidents can cease plant operations and even can cause harm to people, property,

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

and environment. For this reason, to identify vulnerable factors that become unacceptable operating scenarios is a challenge in the risk assessment of complex systems. The risk assessment seeks to minimize undesirable event probability and their impact both for the environment and for the people involved in the operations. The impact in the operation can be measured as economic consequences based on the extension of equipment damage and on reduction of plant performance.

involving complex nonlinearities and multicomponent system, especially, new techniques for risk analysis upon of abnormal event are needed. The quantification of risk cannot be handled with traditional statistical methods since it requires the quantification of the probability of

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The incidents in maritime operations often involve the analysis of low-probability events for which few data are available. Classical statistical methods are inefficient in these cases. Bayesian techniques are useful because of their ability to deal with sparse data and to incorporate a wide variety of information gained based on expert judgment. A further practical advantage of the subjective probability framework in risk assessment applications is that propagation of

In the last few decades, has been several studies examined trends about Bayesian techniques in risk assessment [7–13], such as those presented by Avan and Kvaloy [7] discussing some of the practical challenges of implementing Bayesian thinking and methods in risk analysis, emphasizing the introduction of probability models and parameters and associated uncertainty assessments. Siu and Kelly [8] present a tutorial on Bayesian parameter estimation especially relevant to probability risk assessment. Jun et al. [9] divide the system failure mode based on the criticality analysis using multistage event tree. They predict failure rates and the time to failures and consequently can predict the system reliability. Eleye-Datubo et al. [10] show in a marine evacuation scenario and that of authorized vessels to floating, production, storage, and offloading collision, based on a commercial computer tool. Meel and Seider [11] developed Bayesian model to predict the number of abnormal events in the next time interval utilizing information from previous intervals and determine fuzzy memberships to various critical zones to indicate the proximity of abnormal events to incipient faults, near misses, incidents, and accidents. Kalantarnia et al. [12], for example, use Bayesian theory to update the likelihood of the event occurrence and failure probability of the safety system and hence develop a dynamic failure assessment for a process. Yun et al. [13] use Bayesian estimation for insufficient LNG system failure data; the risk values estimated with these insufficient data may not show statistical stability or represent specific conditions

The quantification of risk requires the quantification of the likelihood of rare accidental events, which normally cannot be done without employing engineering judgment. In this paper the relationship between characteristics and causes of accidents and system components involved in hazardous offloading is analyzed about one type of consequence associated with the incident. This chapter presents a quantitative risk analysis based on Bayesian techniques; the relation between the probability of occurrence of each hazardous event and its consequence could be found; we have developed these concepts in [14]. The objective this approach is providing safety for offloading operations in deepwater oil fields. We consider both FPSO and shuttle as one integrated system. We present the application of risk-based analysis techniques to evaluate offloading operations between a FPSO and a shuttle tanker that could be used to develop actions and procedures to minimize the consequences of an accident for the operation. The methodology presented can provide a model in which reasoning is justified, while it enables a powerful marine decision-support

accidental events that in most cases are rare [7].

of an LNG facility.

uncertainties through complex models is relatively simple.

The search for oil fields no longer occurs exclusively onshore, but includes the oceans of the world. This fact has contributed to the development of rigs for drilling and production offshore in deepwater.

The current method for crude oil export in deepwater is using floating production storage and offloading (FPSO). The FPSO is a floating vessel, in that it is equipped with internal or external turret, and equipment to refine crude oil, and storage capacity. Therefore, FPSO have an offloading system to transfer the crude oil to shuttle tankers. As you can see in [1, 2], the shuttle tankers are increasingly being accepted as a preferred transportation method for remote and deepwater offshore developments, for example, according to ONIP (Programa Nacional de Mobilização da Indústria Nacional do Petróleo e Gás Natural) in 2002, Brazil had 46.0% of the total oil production of Petrobras located in deepwater (400–1000 m) and 29.9% in ultra-deepwater, with water depth greater than 1000 m [3]. More recently, shuttle tankers have become the main way to distribute the crude oil produced offshore on Brazilian fields [4]. The options for methods of offloading from a FPSO and shuttle tanker include remote single point mooring, tandem offloading, and alongside configuration.

The tandem offloading operation is frequently a complex and difficult marine operation. FPSO may rotate due to waves and wind actions, and this rotates according to the weather that generates linear motions of a ship (surge, sway, and yaw). To stay connected for loading and at the same time maintain a safe separation distance, shuttle tanker must position itself aligned with the FPSO position. As we show in [5], the situation is dramatically changed in the tandem offloading operation in terms of positioning complexity and damage potential [5], due to the significant amount of mass involved (e.g., a 150,000-dwt shuttle tanker) in close distance to an installation (FPSO) for a long period of time.

To analyze the nature of the incidents in maritime operations, it is necessary to define a complex relationship among design procedures, equipment, environmental conditions, and operational procedures. To gain a full understanding and comprehensive awareness of safety in each situation, it is necessary to use a systemic approach to consider all the aspects that may lead to hazardous events and to consider different uncertainty sources [6]. In complex system safety assessment, a systemic approach means to consider all functional entities that constitute the system, exploring patterns and inter-relationships within subsystems and seeing undesired events as the products of the working of the system.

In the 1980s and 1990s, the most risk analysts have been trained in the "classical" approach to risk analysis, where probability exists as a quantity characterizing the failure of the system being studied and independent of the analyst. This concept of probability is frequency based, and the results of the risk analyses provide estimates of these "true" probabilities. For operations involving complex nonlinearities and multicomponent system, especially, new techniques for risk analysis upon of abnormal event are needed. The quantification of risk cannot be handled with traditional statistical methods since it requires the quantification of the probability of accidental events that in most cases are rare [7].

and environment. For this reason, to identify vulnerable factors that become unacceptable operating scenarios is a challenge in the risk assessment of complex systems. The risk assessment seeks to minimize undesirable event probability and their impact both for the environment and for the people involved in the operations. The impact in the operation can be measured as economic consequences based on the extension of equipment damage and on

The search for oil fields no longer occurs exclusively onshore, but includes the oceans of the world. This fact has contributed to the development of rigs for drilling and production offshore

The current method for crude oil export in deepwater is using floating production storage and offloading (FPSO). The FPSO is a floating vessel, in that it is equipped with internal or external turret, and equipment to refine crude oil, and storage capacity. Therefore, FPSO have an offloading system to transfer the crude oil to shuttle tankers. As you can see in [1, 2], the shuttle tankers are increasingly being accepted as a preferred transportation method for remote and deepwater offshore developments, for example, according to ONIP (Programa Nacional de Mobilização da Indústria Nacional do Petróleo e Gás Natural) in 2002, Brazil had 46.0% of the total oil production of Petrobras located in deepwater (400–1000 m) and 29.9% in ultra-deepwater, with water depth greater than 1000 m [3]. More recently, shuttle tankers have become the main way to distribute the crude oil produced offshore on Brazilian fields [4]. The options for methods of offloading from a FPSO and shuttle tanker include remote single point

The tandem offloading operation is frequently a complex and difficult marine operation. FPSO may rotate due to waves and wind actions, and this rotates according to the weather that generates linear motions of a ship (surge, sway, and yaw). To stay connected for loading and at the same time maintain a safe separation distance, shuttle tanker must position itself aligned with the FPSO position. As we show in [5], the situation is dramatically changed in the tandem offloading operation in terms of positioning complexity and damage potential [5], due to the significant amount of mass involved (e.g., a 150,000-dwt shuttle tanker) in close distance to an

To analyze the nature of the incidents in maritime operations, it is necessary to define a complex relationship among design procedures, equipment, environmental conditions, and operational procedures. To gain a full understanding and comprehensive awareness of safety in each situation, it is necessary to use a systemic approach to consider all the aspects that may lead to hazardous events and to consider different uncertainty sources [6]. In complex system safety assessment, a systemic approach means to consider all functional entities that constitute the system, exploring patterns and inter-relationships within subsystems and seeing undesired

In the 1980s and 1990s, the most risk analysts have been trained in the "classical" approach to risk analysis, where probability exists as a quantity characterizing the failure of the system being studied and independent of the analyst. This concept of probability is frequency based, and the results of the risk analyses provide estimates of these "true" probabilities. For operations

reduction of plant performance.

106 Probabilistic Modeling in System Engineering

mooring, tandem offloading, and alongside configuration.

installation (FPSO) for a long period of time.

events as the products of the working of the system.

in deepwater.

The incidents in maritime operations often involve the analysis of low-probability events for which few data are available. Classical statistical methods are inefficient in these cases. Bayesian techniques are useful because of their ability to deal with sparse data and to incorporate a wide variety of information gained based on expert judgment. A further practical advantage of the subjective probability framework in risk assessment applications is that propagation of uncertainties through complex models is relatively simple.

In the last few decades, has been several studies examined trends about Bayesian techniques in risk assessment [7–13], such as those presented by Avan and Kvaloy [7] discussing some of the practical challenges of implementing Bayesian thinking and methods in risk analysis, emphasizing the introduction of probability models and parameters and associated uncertainty assessments. Siu and Kelly [8] present a tutorial on Bayesian parameter estimation especially relevant to probability risk assessment. Jun et al. [9] divide the system failure mode based on the criticality analysis using multistage event tree. They predict failure rates and the time to failures and consequently can predict the system reliability. Eleye-Datubo et al. [10] show in a marine evacuation scenario and that of authorized vessels to floating, production, storage, and offloading collision, based on a commercial computer tool. Meel and Seider [11] developed Bayesian model to predict the number of abnormal events in the next time interval utilizing information from previous intervals and determine fuzzy memberships to various critical zones to indicate the proximity of abnormal events to incipient faults, near misses, incidents, and accidents. Kalantarnia et al. [12], for example, use Bayesian theory to update the likelihood of the event occurrence and failure probability of the safety system and hence develop a dynamic failure assessment for a process. Yun et al. [13] use Bayesian estimation for insufficient LNG system failure data; the risk values estimated with these insufficient data may not show statistical stability or represent specific conditions of an LNG facility.

The quantification of risk requires the quantification of the likelihood of rare accidental events, which normally cannot be done without employing engineering judgment. In this paper the relationship between characteristics and causes of accidents and system components involved in hazardous offloading is analyzed about one type of consequence associated with the incident. This chapter presents a quantitative risk analysis based on Bayesian techniques; the relation between the probability of occurrence of each hazardous event and its consequence could be found; we have developed these concepts in [14]. The objective this approach is providing safety for offloading operations in deepwater oil fields. We consider both FPSO and shuttle as one integrated system. We present the application of risk-based analysis techniques to evaluate offloading operations between a FPSO and a shuttle tanker that could be used to develop actions and procedures to minimize the consequences of an accident for the operation. The methodology presented can provide a model in which reasoning is justified, while it enables a powerful marine decision-support

solution that is simple to use, flexible, and appropriate for the risk assessment task. The methodology with Bayesian approach as for decision support is presented in Section 2; we presented the initials theoretically developed in [14], but we include it here again, for the sake of clarity. In Section 3, the application example is presented, and finally, in Section 4 the results and final comments are presented.

#### 2. Dynamic risk assessment methodology

Risk can be represented by Eq. (1) which relates the undesired event's occurrence probability and the consequences:

$$\text{Risk} = \left(p\_i, \mathbf{c}\_i\right)
\text{Risk} = \left(\mathbf{p}\_i, \mathbf{c}\_i\right) \tag{1}$$

where pi is the ith event occurrence probability and ci is the effect of the ith event occurrence [14].

For complex systems, the possibility that an unexpected scenario shows up is related to an initial event or failure which happens in a specific component. For each one of the system's or subsystems' components, it is necessary to know the probabilities that the unexpected condition (failure) shows up, and its consequences and states must be evaluated.

In this context, another important decision-making aspect in complex systems is the need for creating a model which can consider dynamic characteristics of system. In the case under analysis, these characteristics are given by the transition between states corresponding to safe operating zones [15].

Hence, let ST be a variable that represents a state of system, and let K be a scenario. The probability that K be true given the system is in the state ST can be represented by Eq. (2) [5]:

$$P(K|ST) = \frac{P(ST|K) \cdot P(K)}{P(ST)} \tag{2}$$

2.1. Accident modeling

Figure 1. Probabilistic risk assessment methodology.

tion of hazard events [16].

identified [15].

The first step identifies the objective of the risk assessment and to identify and to select the undesirable consequences of interest. These consequences may include items like degrees of harm to environment or degrees of loss of operation. This step covers relevant design and

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109

In this same step, the hazard identification is based on techniques that allow, qualitatively, to assess the consequences of the events of broad impact and to see the effects on the environment, personnel, and facilities. It requires the identification of the hazard event that is one or more physical conditions with the potential to cause damaged. Aiming this stage is to depict the consequences and to determine their causes, because the procedure is based on the selec-

To determine the hazard events, "brainstorming" technique is used involving experienced personnel as well as the procedures used for the practice of routine operations using a question-answer technique based on preliminary hazard analysis (PHA) concepts. Apart from human factors, failures of components installed in complex system are systematically considered by applying the methodology of failure modes and effect analysis, which usually starts from identifying failure modes of each item composing the whole system. Based on information about the system, interviews, and expert opinions, many hazards affecting the system are

operational information including operating emergency procedures.

where P(ST|K) is the probability that the system was in the ST state given a scenario K, P(K) is the probability that a scenario K be true, and P(ST) is the probability that the system is in the state ST.

The method is based on probability risk assessment and Markovian process to aid decisionmaking (see Figure 1). To calculate the probability of accident scenario, the Bayesian approach is presented in detail in [5]. It is used to estimate the probabilities that the system is in each state stochastic model are applied. This methodology allows, quantitatively, to assess the consequences of the events of broad impact and to see relationship between the environment changes and those impacts. The methodology can be summarized in four steps: accident modeling, failure probability assessment with Bayesian techniques, evaluation of consequences, and Markovian process to aid decision-making.

Decision-Making Model for Offshore Offloading Operations Based on Probabilistic Risk Assessment http://dx.doi.org/10.5772/intechopen.75833 109

Figure 1. Probabilistic risk assessment methodology.

#### 2.1. Accident modeling

solution that is simple to use, flexible, and appropriate for the risk assessment task. The methodology with Bayesian approach as for decision support is presented in Section 2; we presented the initials theoretically developed in [14], but we include it here again, for the sake of clarity. In Section 3, the application example is presented, and finally, in Section 4

Risk can be represented by Eq. (1) which relates the undesired event's occurrence probability

where pi is the ith event occurrence probability and ci is the effect of the ith event occu-

For complex systems, the possibility that an unexpected scenario shows up is related to an initial event or failure which happens in a specific component. For each one of the system's or subsystems' components, it is necessary to know the probabilities that the unexpected condi-

In this context, another important decision-making aspect in complex systems is the need for creating a model which can consider dynamic characteristics of system. In the case under analysis, these characteristics are given by the transition between states corresponding to safe

Hence, let ST be a variable that represents a state of system, and let K be a scenario. The probability that K be true given the system is in the state ST can be represented by Eq. (2) [5]:

P Kð Þ¼ <sup>j</sup>ST P ST ð Þ <sup>j</sup><sup>K</sup> <sup>∙</sup>P Kð Þ

where P(ST|K) is the probability that the system was in the ST state given a scenario K, P(K) is the probability that a scenario K be true, and P(ST) is the probability that the system is in the

The method is based on probability risk assessment and Markovian process to aid decisionmaking (see Figure 1). To calculate the probability of accident scenario, the Bayesian approach is presented in detail in [5]. It is used to estimate the probabilities that the system is in each state stochastic model are applied. This methodology allows, quantitatively, to assess the consequences of the events of broad impact and to see relationship between the environment changes and those impacts. The methodology can be summarized in four steps: accident modeling, failure probability assessment with Bayesian techniques, evaluation of conse-

quences, and Markovian process to aid decision-making.

; ci

(1)

P ST ð Þ (2)

; ci Risk <sup>¼</sup> pi

Risk ¼ pi

tion (failure) shows up, and its consequences and states must be evaluated.

the results and final comments are presented.

108 Probabilistic Modeling in System Engineering

and the consequences:

operating zones [15].

state ST.

rrence [14].

2. Dynamic risk assessment methodology

The first step identifies the objective of the risk assessment and to identify and to select the undesirable consequences of interest. These consequences may include items like degrees of harm to environment or degrees of loss of operation. This step covers relevant design and operational information including operating emergency procedures.

In this same step, the hazard identification is based on techniques that allow, qualitatively, to assess the consequences of the events of broad impact and to see the effects on the environment, personnel, and facilities. It requires the identification of the hazard event that is one or more physical conditions with the potential to cause damaged. Aiming this stage is to depict the consequences and to determine their causes, because the procedure is based on the selection of hazard events [16].

To determine the hazard events, "brainstorming" technique is used involving experienced personnel as well as the procedures used for the practice of routine operations using a question-answer technique based on preliminary hazard analysis (PHA) concepts. Apart from human factors, failures of components installed in complex system are systematically considered by applying the methodology of failure modes and effect analysis, which usually starts from identifying failure modes of each item composing the whole system. Based on information about the system, interviews, and expert opinions, many hazards affecting the system are identified [15].

The accident modeling is finished with scenario modeling based on the use of the event tree. An event tree is used to identify the various paths that the system could take, starting with the initiating event and studying the failure progress as a series of successes or failures of intermediate events called hazard events, until an end state is reached. That sequence of events is named failure scenario for which the consequences are estimated.

instance, time to failure, the exponential distribution is the proper likelihood [8]. However, situations can arise where more complicated likelihood functions need to be constructed. Given a process model, general approaches for developing functions of random variables can be used

Decision-Making Model for Offshore Offloading Operations Based on Probabilistic Risk Assessment

Prior distributions can be specified in different forms depending on the type and source of information as well as the nature of the random variable of interest. The prior distributions can be informative prior distributions when it is one that reflects the analyst's beliefs concerning an unknown parameter or noninformative prior distributions when large amounts of data are available and when the analyst's prior beliefs are relatively vague. This paper deals with informative prior distributions deals. When it is assumed that the prior is a member of some parametric family of distributions, the form can be parametric and numerical. Among the parametric form are the gamma or lognormal for rates of events and beta for event probabilities per demand. Bayesian statistics combines knowledge about the parameter, which is reflected by the prior distribution, and information from the data, which is contained in the likelihood function. Using Bayes' theorem in its continuous form, the prior probability distribution of a continuous unknown quantity, P0(x), can be updated to incorporate new evidence

> P xð Þ¼ <sup>j</sup><sup>E</sup> L Eð Þ <sup>j</sup><sup>x</sup> <sup>∙</sup>P0ð Þ<sup>x</sup> Ð

where P(x|E) is the posterior probability distribution of the unknown quantity x given evi-

For some combinations of likelihood functions and prior distributions, Eq. (3) must be evaluated numerically. For a given model, there is a family of distributions where if the prior distribution is a member of this family, then the posterior distribution will be a member of the same family. These families of distribution are called conjugate distribution [19]. The conjugate likelihood and prior are most commonly used in probability risk assessment as well as the form of the resulting posterior distributions. These combinations are

Prior P0(x) Likelihood L(E|x) Posterior P(x|E)

Poisson (x) ð Þ <sup>x</sup>∙<sup>t</sup> <sup>r</sup> <sup>r</sup>! <sup>e</sup>�x∙<sup>t</sup>

Poisson (x) ð Þ <sup>x</sup>∙<sup>t</sup> <sup>r</sup> <sup>r</sup>! <sup>e</sup>�x∙<sup>t</sup>

Binomial (r, n) n! <sup>r</sup>!ð Þ <sup>n</sup>�<sup>r</sup> ! <sup>x</sup><sup>r</sup>

ð Þ <sup>1</sup> � <sup>x</sup> <sup>n</sup>�<sup>r</sup>

L Eð Þ <sup>j</sup><sup>x</sup> <sup>∙</sup>P0ð Þ<sup>x</sup> <sup>∙</sup>dx (3)

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111

Beta (α,β) <sup>Γ</sup>ð Þ <sup>α</sup>þ<sup>β</sup>

Numerical

xα0�<sup>1</sup> <sup>Γ</sup> <sup>α</sup><sup>0</sup> ð Þ <sup>e</sup>�β<sup>0</sup> ∙x

<sup>Γ</sup>ð Þ <sup>α</sup> <sup>∙</sup>Γð Þ <sup>α</sup> <sup>∙</sup>x<sup>α</sup>�<sup>1</sup>∙ð Þ <sup>1</sup> � <sup>x</sup> <sup>β</sup>�<sup>1</sup>

Gamma (α' = α + r, β' = β + t)

to develop likelihood functions [18].

E, as shown in Eq. (3):

shown in Table 1.

<sup>Γ</sup>ð Þ <sup>α</sup> <sup>∙</sup><sup>Γ</sup>ð Þ<sup>β</sup> <sup>∙</sup>x<sup>α</sup>�<sup>1</sup>∙ð Þ <sup>1</sup> � <sup>x</sup> <sup>β</sup>�<sup>1</sup>

Beta (α,β) <sup>Γ</sup>ð Þ <sup>α</sup>þ<sup>β</sup>

Gamma (α,β) xα�<sup>1</sup> <sup>Γ</sup>ð Þ <sup>α</sup> <sup>e</sup>�β∙<sup>x</sup>

Lognormal (μ,σ)

1 ffiffiffiffiffi <sup>2</sup>∙<sup>π</sup> <sup>p</sup> <sup>∙</sup>σ∙<sup>x</sup> e �1 2 lnx�<sup>μ</sup> ð Þ <sup>σ</sup>

dence E and L(E|x) is the likelihood function.

Table 1. Typical prior and likelihood functions [19].

### 2.2. Failure probability assessment

In this step the failure probability of occurrence of a failure scenario is calculated combining two conventional reliability analysis methods: fault tree analysis (FTA) and event tree.

The probability of each failure scenario is determined by summing the probability of each set of events which lead to this outcome. Each sequence probability is obtained by simply multiplying the probabilities of the events represented in each branch of the event tree in the case of independence case; if there is dependence between events, the Bayesian methods are used. The probabilities of the hazard event are obtained by solution of fault trees associated with each hazard event. Fault tree analysis is a systematic, deductive, and probabilistic risk assessment tool which clarifies the causal relations leading to a given undesired event. A fault tree is quantified considering that its basic events tend to follow a probability distribution. The failure probability of basic events is calculated using Bayesian methods.

### 2.2.1. Bayesian ideas and data analysis

The Bayesian techniques are appropriate for use in offshore offloading operation analysis because the Bayesian statistical analysis involves the explicit use of subjective information provided by the expert judgment, since initial uncertainty about unknown parameters of failure distribution of basic events must be modeled from a priori expert opinion or based on insufficient data and evidence collected. Bayes' theorem has been proven to be a powerful coherent method for probabilistically processing new data, as they become available over time, so that the current posterior distribution can then be used as the prior distribution when the next set of data becomes available.

The Bayesian method starts identifying the parameter to be estimated. This involves the consideration of the form of the likelihood function appropriate to the evidence that will be collected. The second step is development of prior probabilities to describe the system current state of knowledge. The next step incorporates information through the collection of evidence and construction of the likelihood function selected in the stage one. The final step results in new probabilities using Bayes' theorem, called posterior distribution, to describe your state of knowledge after combining the prior probabilities with the evidence [17].

The selection of an appropriate likelihood function requires engineering knowledge specific to the process being modeled, as well as the way the new data or evidences are generated. When modeling the number of failures associated with a given piece of equipment, the Poisson distribution is the proper likelihood function. While when modeling the number of failures on system demands, the binomial distribution is the proper likelihood function. For data in form of expert judgment, lognormal distribution is a proper likelihood function. For continuous data, for instance, time to failure, the exponential distribution is the proper likelihood [8]. However, situations can arise where more complicated likelihood functions need to be constructed. Given a process model, general approaches for developing functions of random variables can be used to develop likelihood functions [18].

Prior distributions can be specified in different forms depending on the type and source of information as well as the nature of the random variable of interest. The prior distributions can be informative prior distributions when it is one that reflects the analyst's beliefs concerning an unknown parameter or noninformative prior distributions when large amounts of data are available and when the analyst's prior beliefs are relatively vague. This paper deals with informative prior distributions deals. When it is assumed that the prior is a member of some parametric family of distributions, the form can be parametric and numerical. Among the parametric form are the gamma or lognormal for rates of events and beta for event probabilities per demand. Bayesian statistics combines knowledge about the parameter, which is reflected by the prior distribution, and information from the data, which is contained in the likelihood function. Using Bayes' theorem in its continuous form, the prior probability distribution of a continuous unknown quantity, P0(x), can be updated to incorporate new evidence E, as shown in Eq. (3):

$$P(\mathbf{x}|E) = \frac{L(E|\mathbf{x}) \cdot P\_0(\mathbf{x})}{\int L(E|\mathbf{x}) \cdot P\_0(\mathbf{x}) \cdot d\mathbf{x}} \tag{3}$$

where P(x|E) is the posterior probability distribution of the unknown quantity x given evidence E and L(E|x) is the likelihood function.

For some combinations of likelihood functions and prior distributions, Eq. (3) must be evaluated numerically. For a given model, there is a family of distributions where if the prior distribution is a member of this family, then the posterior distribution will be a member of the same family. These families of distribution are called conjugate distribution [19]. The conjugate likelihood and prior are most commonly used in probability risk assessment as well as the form of the resulting posterior distributions. These combinations are shown in Table 1.


Table 1. Typical prior and likelihood functions [19].

The accident modeling is finished with scenario modeling based on the use of the event tree. An event tree is used to identify the various paths that the system could take, starting with the initiating event and studying the failure progress as a series of successes or failures of intermediate events called hazard events, until an end state is reached. That sequence of events is

In this step the failure probability of occurrence of a failure scenario is calculated combining

The probability of each failure scenario is determined by summing the probability of each set of events which lead to this outcome. Each sequence probability is obtained by simply multiplying the probabilities of the events represented in each branch of the event tree in the case of independence case; if there is dependence between events, the Bayesian methods are used. The probabilities of the hazard event are obtained by solution of fault trees associated with each hazard event. Fault tree analysis is a systematic, deductive, and probabilistic risk assessment tool which clarifies the causal relations leading to a given undesired event. A fault tree is quantified considering that its basic events tend to follow a probability distribution. The failure

The Bayesian techniques are appropriate for use in offshore offloading operation analysis because the Bayesian statistical analysis involves the explicit use of subjective information provided by the expert judgment, since initial uncertainty about unknown parameters of failure distribution of basic events must be modeled from a priori expert opinion or based on insufficient data and evidence collected. Bayes' theorem has been proven to be a powerful coherent method for probabilistically processing new data, as they become available over time, so that the current posterior distribution can then be used as the prior distribution when the

The Bayesian method starts identifying the parameter to be estimated. This involves the consideration of the form of the likelihood function appropriate to the evidence that will be collected. The second step is development of prior probabilities to describe the system current state of knowledge. The next step incorporates information through the collection of evidence and construction of the likelihood function selected in the stage one. The final step results in new probabilities using Bayes' theorem, called posterior distribution, to describe your state of

The selection of an appropriate likelihood function requires engineering knowledge specific to the process being modeled, as well as the way the new data or evidences are generated. When modeling the number of failures associated with a given piece of equipment, the Poisson distribution is the proper likelihood function. While when modeling the number of failures on system demands, the binomial distribution is the proper likelihood function. For data in form of expert judgment, lognormal distribution is a proper likelihood function. For continuous data, for

knowledge after combining the prior probabilities with the evidence [17].

two conventional reliability analysis methods: fault tree analysis (FTA) and event tree.

named failure scenario for which the consequences are estimated.

probability of basic events is calculated using Bayesian methods.

2.2. Failure probability assessment

110 Probabilistic Modeling in System Engineering

2.2.1. Bayesian ideas and data analysis

next set of data becomes available.
