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The f-N diagram is used as a means of presenting information about societal risks. In Norsok Standard Z -013, f-N curve is defined as a risk parameter that the curve represents the frequency (f) of accidents causing N-fatalities  .
Ball, 1998 cited in the Guidelines for Developing Quantitative Safety Risk Criteria  states that f-N curve and F-N diagrams may be used to depict at least three different types of information:
The historical record of incidents
The results of a quantitative risk analysis (QRA)
Criteria for judging the tolerability of risk
To calculate values for total risk for the installation and the presentation in plot the values in an f-N diagram, a list of all of the events (Accident Categories) and their associated frequencies and consequences (number of fatalities, N) will be compiled and sorted by decreasing value of N (Number of fatalities). Given Table 1 with the fatality risk results for an example installation;
Table : Fatality risk results for an example installation
In order to understand the contribution of all different types of accidents to risk potential, Potential Loss of Life is first plotted. Figure 1 shows that the most exposed area consists of the Process leaks whereby the PLL are about 2.655 10-2. This is in confirmation with many cases of the offshore installation. Most frequent leaks happen in the process area due to the gas leakage which leads generally to ignition and further to an explosion or a fire which generally ends to high fatality.
Figure : Risk contributions to PLL
Vinnem, 2007 defines the Potential Loss of Life (PLL) as the expected number of fatalities per year and can be formulated as follows :
fnj annual frequency of accident scenario (event tree terminal event) n with personnel consequence j
cnj expected number of fatalities for accident scenario (event tree terminal event) n with personnel consequence j
n total number of accident scenarios (event tree terminal event) in all event trees
j total personnel consequences types, usually immediate, escape, evacuation and rescue facility
In Table 2, process leaks are the most severe and has a predicted consequence of 2 to 5 fatalities, which occurs at a predicted frequency of 3.90 10-3. The second most severe event is the occupational accidents, which has a predicted consequence of 21 to 50 fatalities, occurs at a predicted frequency of 4.20 10-3. It is followed by Riser/pipelines leaks.
Table : Table of frequency of hazards
Table 3 presents the F-N QRA results. Rearranging the data from table 2 in decreasing order of the number of fatalities gives the data set presented in Table 3. Values presented in each row in the table 3 calculate the cumulative frequency of occurrence for that event and all events preceding it in the table. To put this data into the form needed for plotting an F-N curve, the frequency must be expressed in terms of the cumulative frequency. The resulting data are plotted as an F-N curve in Figure 2.
Table : F-N Presentation of QRA Results
Many analysts believe that societal risk results are most effectively presented in graphical form. Such plots are normally log-log plots with the x-axis representing the consequences and the y-axis representing the frequency of occurrence. Log-log scales are used because the range of values for f and N can span multiple orders of magnitude.
By definition Societal risk (SR) is the relationship between the frequency and the number of people suffering from a specified level of harm in a given population from the realization of specified hazards, Institution of Chemical Engineers 1992 .
Another way to communicate risk criteria will be to look into the significance of the slope of the F-N curve. When the F-N curve slope is equal to -1, the risk criterion is termed 'risk neutral.' In this case, the risk criterion would dictate that the frequency of an event that results in 100 or more fatalities must be 10 times lower than the frequency of an event that results in 10 or more fatalities.
Figure : F-N Curve for Hypothetical QRA Results
Guidelines suggest two ways of plotting societal risks:
One fashion known as Non-cumulative frequency basis. For these graphs, called f-N curves, the value plotted on the y-axis is the discrete frequency of experiencing exactly N fatalities.
The second fashion is the Cumulative frequency basis. Here graphs, called F-N curves, the value plotted on the y-axis is the cumulative frequency of experiencing N or more fatalities.
Mathematically, the equation for an F-N criterion curve may be presented as: [Ball 1998]
F = the cumulative frequency of N or more fatalities
N = the number of fatalities
a = aversion factor (often between 1 and 2)
k = constant
In Norwegian Regulatory Requirement, it has been a trend of an increasing focus on the ALARP principle compared to Risk Acceptance criteria wide used before 2004 . The ALARP principle implies that all risk reduction proposals that are well founded should be implemented unless it may be shown that costs and/or other negative effects are in gross disproportion to the benefits .
Using guidelines given by , we obtain the Hypothetical QRA Results with EU Societal Risk Criterion. It demonstrated that the incidence of high fatalities in the studied system is negligible while few accidents can happen often but with low fatalities.
Figure : Hypothetical QRA Results with EU Societal Risk Criterion
Based on the Advisory Committee on Major Hazards, 1976 a serious accident in a particular plant was unlikely to occur more often than once in 10,000 years, which could be regarded on the border of acceptability, ACMH 1976. This has often been taken as an anchor point for the F-N curve where the chance of an accident involving 10 or more fatalities should not exceed 1 in 10,000 per year . In the second Canvey report, HSE 1981, it was suggested that an event with 1,500 fatalities and the frequency of 2 x 10-4 per year (2 in 10,000) could be judged as intolerable .
Discussion of sustainability/justifiability of present employee risk levels in Norwegian sector (Part C)
In the section, it is discussed whether the risk levels for employees in the Norwegian petroleum sector are at levels that are sustainable/justifiable in a societal context, and it is also identified areas in a need for further improvements in the offshore operations.
The basis of our discussion is the paper: "On the risk to personnel in the offshore industry" by J.E Vinnem, 2008. Vinnem uses various sources including information from the Risk Level Project  to categorize components of the fatality risk picture in occupational accidents, major accidents on the installation, and transportation accidents during transfer between shore and offshore installations. From here he gives predictions of what the current risk levels are .
From quantitatively as well as qualitatively perspectives, Vinnem makes the most relevant predictions about occurrence of fatalities in the Norwegian national sector for the period 2008-2012. In his study, J.E. Vinnem, 2008 uses Bayesian approach which implies that the risk values express our belief (uncertainty) about what will occur in the future in terms of accidents and consequences. Thus sensitivity studies have been used extensively in order to explore the sensitivity of the risk model to variations in data and assumptions.
Through performed assessment and based on predicted number of fatalities in the Norwegian Continental Shelf, and based on analysis the causes of fatalities as well as the variation of fatalities between 1985 and 2005, J. E. Vinnem came to a conclusion that there are indications that trends of fatal accident occurrence become few and rare. The relative distribution of the risk level from major hazards has almost reduced up to 50% by 2008.
This section is divided into three parts: 1) the key findings of the paper "On the risk to personnel in the offshore industry", 2) lessons learned and trends in fatalities rates and 3) the improvement needed for sustainable employee risk level in Norwegian sector.
Fatal accident rates in worldwide offshore operations
Based on the OGP Annual statistical summaries, Vinnem demonstrates a general decrease of fatal accidents rate worldwide for the period 1997 and 2006 with the highest number of fatalities of 32 loss of life in 2004. The lowest registered is 10 lives, in 2005 . The high value of fatalities found in 2004 is due to two helicopter accidents that occurred in 2004 .
 found that not all the relevant accidents are included in the OGP report. As an example he pointed out that the Risk Level Project report documents ten fatalities in offshore accidents in the UK in the period 2001-06, when the OGP report considers for the same period, seven fatalities offshore . Others accidents not included in the OGP are the capsize of P-36 offshore Brazil in 2001 followed by an explosion and fire which resulted in 11 fatalities; a burning blowout occurred offshore Egypt, without fatalities in 2004.
Meanwhile Vinnem noticed that the estimated FAR values will not be significantly affected if it is considered an increase of volume of exposure hours of 694 million manhours from 39 companies in 75 countries by 2006 .
The case of Norwegian offshore operations
In Norwegian sector, major accidents occurred quite frequently during the first 20 years of operations in the Norwegian sector, but from 1985 to 2006 their incidence got declined. The last ignited hydrocarbon leak on an installation in the Norwegian sector was in November 1992 [cited in ]. Due to the absence of major accidents between 1985-2006,  argues that the likelihood of major accidents in the Norwegian sector is reduced extensively.
An average FAR calculated in  and normalized against exposure hours in the period 1974-2006 and 1967-2006 shows 0.66 fatalities per 100 million exposure hours (1974-2006) for production installations and 45.4 fatalities per 100 million exposure hours (1967-2006) for mobile installations.  points out that the high value of fatalities for mobile installations is linked to the capsize of mobile accommodation unit Alexander Kielland, 1980, with 123 fatalities . A best way to calculate an average FAR without the influence of the very high number of fatalities like the one in the Alexander Kielland accident,  suggests to calculate an occurrence frequency for major accidents, based on history. In this case the expected number of fatalities per major accident in the future are derived in  as 0.25 major accidents per 100 million exposure hours (1974-2006) for the production installations, while for the mobile installations, the value becomes 1.10 major accidents per 100 million exposure hours (1967-2006).
Since major accidents have not occurred for many years,  performs the calculation of the frequency of major accidents first for a considered period 1980-2006 and then for the period 1990-2006. The results obtained are compared with other alternative approach such QRA that consider the predicted frequency and fatalities for individual installations in the future.
The table below shows a summary of fatalities per 100 million exposure hours for the different periods as computed by J.E Vinnem .
Table : Computed fatalities per 100 million exposure hours based on J.E. Vinnem, 2008, source 
Fatalities per 100 million exposure hours
Assumed expected 8 fatalities per major accident in the future
QRA studies, and assumed expected 5 fatalities for individual installations in the future by J.E. Vinnem
Expert evaluations &judgment
(refined over years) with input from the Risk Level Project 
Average assumed for the period 1990-2006 and the
basis for the values in the Risk Level Project 
The numbers of fatalities obtained were based on assumption of expected potential loss of life in the future per major accident for production installations as well as for mobile installations.
From the table above,  found that the various predictions of the fatality numbers for production installations were reasonably consistent; implying that the uncertainty associated with this prediction should be limited. However the predictions for mobile installations were not quite as consistent .
For the category of helicopter transport in Norwegian sector, the conducted analysis by  shows that average fatal Accident Rate for the period 1997-2006 was estimate at 1.89 10-6 fatalities per person flight hours while for the period 1987-2006 the average fatal Accident Rate has turned out to be 1.15 10-6 fatalities per person flight hours.
The prediction of future risk level.
To predict the future risk level  uses Bayesian approach for five year period from 2008 to 2012. The assumption taken was that the average fatal accident during the last 10 years is the most applicable for the future five years.
Another important principle adopted in the Vinnem's paper is the triangulation, the same approach was used in the Norwegian Risk Level Project, 1998 cited in  to predict the trends of risk level for the period 1998-2008. This was implemented through input from various sources and perspectives, quantitatively as well as qualitatively, including information from the Risk Level Project.
For different categories, separate assumptions were made to appropriately account the complexity of each system to be analyzed. In the table 5 is given the estimated fatalities for each category.
Table : Assumed fatalities per 100 million exposure hours based on J.E. Vinnem, 2008, source 
Fatalities per 100 million exposure hours
Major Accidents on Installations
The assumption for the occupational accidents is based on a long period registered without occupational fatalities. The last occupational fatality on mobile units in the Norwegian sector was on 13.12.1993 . While the last fatalities on production and mobile installations were in 2002, in November and April, 2002 .
Regarding the Major Accidents on Installations, Vinnem supplements the no occurrences of fatal accident in the period 1990-2006, with data from QRA studies and the Risk Level Project. It followed that various predictions of the fatality numbers observed for production installations were reasonably consistent which was a confirmation that the uncertainty associated with this prediction would be limited .
For Helicopter Transportation Accidents,  takes into account the effect of recent improvements and the new introduced Sikorsky and Super Puma models for personal offshore transportation. The paper also considers the goal of the White Paper guidelines with a target to reduce by 50% of the fatality frequency. All those measures combined to stipulated actions by the helicopter safety, (that include Flight data monitoring, New technology, TCAS 1 collision avoidance system, EGPWS, Enhanced Ground Proximity Warning System, De icing (rotor), Survivability in Sea state 6), Vinnem demonstrates the taken actions for fleet modernization will definitely lower the risk for major helicopter accidents in the future.
On this basis, the paper "On the risk to personnel in the offshore industry" by J.E Vinnem, 2008 has subjectively considered that a representative average value for the period 2008-2012 may be 90 fatalities per 100 million person flight hours for Helicopter transport. The suggested value corresponds to 70% as an average achievement of the 50% reduction which was established as the goal for the period 2002-2012 .
In table 6 is summarized the predicted risk level for the period 2008-2012 based on Vinnem, 2008 :
Table : Predicted Future Fatalities - Norwegian Sector 2008-2012 based on J.E. Vinnem, 2008 
Fatalities per 100 million exposure hours
Major Accidents on Installations
Helicopter Transportation Accidents
The values obtained from preceding table were used by  to finally predict the expected number of fatal accidents in the period 2008-2012 for the entire Norwegian Continental Shelf. 2.4 fatal accidents were predicted for the occupational accidents; 0.26 for the helicopter accidents; while 0.35 fatal accidents were predicted for the major accidents on installations.
In order to illustrate the uncertainty of the predictions  has applied the following prediction intervals for the number of fatal accidents in the five year period 2008-2012:
Occupational accidents: 0-3 fatal accidents
Helicopter accidents: 0-2 fatal accidents
Major accidents: 0-2 fatal accidents
Lessons learned and Trends in Fatality Rates
The paper "On the risk to personnel in the offshore industry" by J.E Vinnem, 2008 and the study of the "Risk Level in Norwegian Petroleum Activities: Summary report" by PSA, 2012 shows that there has been a significant reduction of the number of fatal accidents since the inception of the "Trends in risk level - Norwegian Continental Shelf" project in 1999/2000 [6, 7].
Within PSA's area of responsibility offshore and on land, there were no fatal accidents occurred during 2011. Five people have died in occupational accidents over the past 10 years . For production installations there is a downward trend over the entire period, the level was stable until 2000 and the frequency has been falling since year 2000.
The risk level project has set 21 defined hazard and accident situations (DFUs) to describe the development in risk levels. Based on DFU, there have been no major accidents, according to definition, on facilities on the Norwegian Continental Shelf after 1990. None of the DFUs that indicate major accident risk on facilities have resulted in fatalities during the period. The last time there were fatalities in connection with one of these major accident's DFUs was in 1985, with the shallow gas blowout on the mobile facility "West Vanguard [7, 8]
However a fatal accident occurred at a land-based plant in 2005 ; in 2009 there was one fatal accident within the PSA' s area of authority on the Norwegian Continental Shelf, on Oseberg B, 7th May 2009, during dismantling of scaffolding .
Outside the NCS, there have been several serious helicopter accidents associated with petroleum activities in 2009. On 12th March there were 17 fatalities when a Sikorsky S-92 ditched into the sea off Newfoundland and on 1st April 16 persons lost their lives when a Super Puma L2 helicopter ditched off the Scottish coast On 18th February the pilots of a Super Puma EC-225 performed a controlled landing in the sea on the ETA P Field off Scotland. All 18 people on board were rescued .
In Norway, a controlled emergency landing was performed with a Sikorsky S-92 on Tor on 8th April .
Traditionally, the industry has selected indicators to illustrate safety trends in petroleum activities based on the frequency of occupational accidents resulting in lost working time. In recent years it has been acknowledged that the trends in risk levels in the petroleum industry are not only a matter of concern to everyone involved in the industry but are also of interest to the public at large . Today, the preference has been for a range of indicators to be used to measure trends in certain key HES factors  to give a wide range picture of the safety situation.
In this study we found that the outcomes of the Vinnem's paper: "On the risk to personnel in the offshore industry" in term of fatalities prediction are an agreement with the statistics records of PSA reports trough the RNNS project for the period 2008-2012. This is a confirmation that the applied Bayesian approach combined with qualitative and quantitative methods to predict trend risk level have yielded satisfactory results.
The observed decrease of fatal accident and fatalities rate in the NCS, is also experienced for the case of personal injuries recorded in the offshore operation. Serious personal injuries have shown a favorable trend in recent years, reaching 0.6 per million hours worked for the whole NCS in 2011 - significantly below the average for the preceding 10 years . Even with more hours worked overall, serious injuries fell from 23 in 2010 to 17 in 2011.
The figure 4 shows the trends of serious personal injuries on mobile and production facilities in relation to working hours. On mobile units, the personal injury frequency constantly decreased from 2.9 to 1.58 serious personal injuries per million working hours for the period 2001-2004. For the production the same situation is observed with a decrease from 1.89 to 0.79 for the same period.
From 2005 to 2008, the situation remained more stable whereby the number of serious personnel injuries varied around 1.77 and 1.25 for mobile facilities and 1.1 and 0.65 for production facilities. From 2009 to 2011, the number of serious personnel injuries on mobile has known a significant decrease with 36% of decrease of serious personnel injuries in 2009 compared to previous year. For production there was not significant change but from 2010 it was nevertheless below the average for the preceding decade. Nine such injuries were recorded in 2011, compared with five the year before .
Figure : Serious personal injuries on mobile and production facilities related to working hours
The frequency of severe personnel injury on production and mobile installations, as reported in the figure 4, shows pick in early year of 2001 while for the following years au to 2011, there has been a considerable decline of injury frequency on mobile as well as on productions facilities. On mobile facilities in 2011 shows an increase in the frequency for serious personal injuries from 0.4 in 2010 to 0.7 in 2011. The injury frequency is just below the average for the ten previous years 
It should be noted that the number of manhours on production installations is typically 3-4 times that on mobile installations. It should also be noted that two fatalities have occurred on a crane vessel during periods when it was operating in the Norwegian sector, in 2003 and 2007 .
To what extent further improvements are needed?
Though major accidents have not occurred in the Norwegian sector after 1985, there were cases of near misses. Two near-misses have occurred in the Norwegian sector during the last few years, the unignited subsea blowout on Snorre Alpha in November 2004, and the massive unignited gas leak on the Visund FPU in January 2006 .
During the period 2003 - 2010, a total of 146 well control events were reported on the Norwegian Continental Shelf, of which only about ten events have been investigated . Since the major accident potential inherent in well control events is indisputable, it is of utmost importance that more investigations of such event are made. This will provide the necessary insight in causal mechanisms, complex connections and framework conditions that contribute to such events, which in turn is a precondition for efficient measures and experience transfer in the industry. Increased efforts directed toward barrier management, more investigations of well control events and operational risk assessments, will be instruments to ensure understanding of major accident risk in connection with well control events .
On the other hand we could argue that as long as major accidents have not occurred for many years, it is a good sign of improved mechanisms put in place by the Norwegian policy and the industry for risk management as well as robust barrier to prevent incidences happening or developping. Meanwhile in the average over the entire production period for the Norwegian sector, it is unknown the actual extent of these improvements.
The PSA has adopted a set of main priorities with focus on management, technical and operational barriers, preventing environmental harm and groups particularly exposed to risk 
Based on information obtained for J.E Vinnem paper and the use of PSA's published reports, below is suggested areas needed for improvement. The list may be long but here we use few that were judged the most important.
This emphasizes the importance of making standards such as NORSOK D-001 and D-010 more aggressive as regards stipulating requirements that contribute to continuous improvement. Following the blowout and a catastrophic explosion on the Deepwater Horizon offshore oil drilling platform April 20th, 2010 it has been an increased focus on better technical solutions associated with systems for detecting well kicks, presentation of safety-critical information for the driller and drilling fluid logger, design of the drilling cabin and systems/technology for pore pressure predictions .
According to Magne Ognedal, director-general in the Petroleum Safety Authority Norway; in order to work purposefully on continuous safety improvements, "the information will form a key part of the basis for planning our activities on possible changes to Norway's regulations and with supervision".
The barrier indicator related to major accidents shows that a relatively large number of installations had fairly substantial non-conformities from the expected industry level. This means that the sector has a clear improvement potential with regard to ensuring that these barriers are sufficiently robust .
The PSA has responsibility to monitor the way company managements work to reduce major accident risk over a number of years. PSA audit results, experience from major accidents nationally and internationally and recognised accident theory indicate that management plays a key role for major accident risk.
Initiatives and decisions taken by management define and influence conditions which are of significance for such threats.
The PSA will focus particular attention in 2012 on internal follow-up by the companies and the overview they have of their own operations. Follow-up, including of contractors, is a key tool for exercising management responsibility in this area.
Much attention will also be devoted by the PSA to the responsibility of drilling contractors, how they understand and exercise this responsibility, and how management in the companies works to reduce major accident risk.
Groups exposed to risk
In the section 4.3., it has been reported a high level of serious personal injuries. This constitutes a threat for a certain group of employee exposed to risk. Specific measures should be adopted by companies to reduce the threat of injury and illness for groups particularly exposed to such risks.
According to PSA, the risk of occupational injury and illness fall unequally on different categories of workers in the petroleum industry. Groups of contractor employees have more risk factors in their working environment, and their exposure is higher than for operator personnel . Among the priorities should be the cutting noise levels to prevent hearing damage, measures to reduce the high cost of occupational illness and injury
The paper "On the risk to personnel in the offshore industry" by J.E Vinnem, 2008 has shown that that accident statistics from the past may be used in order to predict the number of accidents and fatalities in the future.
Due to the absence of major accidents after 1985, it might be argued that the likelihood of major accidents in the Norwegian sector is reduced extensively. But there have been recent near misses in the Norwegian sector as noted above, and there have been major accidents in the UK sector as well as in more distant areas. A balanced evaluation is therefore needed.
The risk levels reduction registered with operation of production installation, mobile units and the helicopter transportation in the Norwegian sector implies that considerable improvement has been made in all three main categories.
To reach 'zero vision' which is adopted across the industry as well as for sustainable employee risk level in Norwegian sector a continuous improvement towards will be needed in all three areas.