Article pubs.acs.org/est
Statistics of Extremes in Oil Spill Risk Analysis Zhen-Gang Ji,* Walter R. Johnson, and Geoffrey L. Wikel Bureau of Ocean Energy Management, 381 Elden Street, Herndon, Virginia 20170, United States ABSTRACT: The Deepwater Horizon oil spill (DWH) in 2010 in the Gulf of Mexico is the largest accidental marine oil spill in the history of the petroleum industry. After DWH, key questions were asked: What is the likelihood that a similar catastrophic oil spill (with a volume over 1 million barrels) will happen again? Is DWH an extreme event or will it happen frequently in the future? The extreme value theory (EVT) has been widely used in studying rare events, including damage from hurricanes, stock market crashes, insurance claims, flooding, and earthquakes. In this paper, the EVT is applied to analyze oil spills in the U.S. outer continental shelf (OCS). Incorporating the 49 years (1964−2012) of OCS oil spill data, the EVT is capable of describing the oil spills reasonably well. The return period of a catastrophic oil spill in OCS areas is estimated to be 165 years, with a 95% confidence interval between 41 years and more than 500 years. Sensitivity tests indicate that the EVT results are relatively stable. The results of this study are very useful for oil spill risk assessment, contingency planning, and environmental impact statements on oil exploration, development, and production.
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INTRODUCTION The Deepwater Horizon oil spill (DWH) began on April 20, 2010 in the Gulf of Mexico on the BP’s Macondo Prospect. The explosion and sinking of the Deepwater Horizon oil rig killed 11 workers. The exploration well flowed for 87 days, until it was capped on July 15, 2010. The total oil discharge is estimated at 4.9 million barrels1 (MMbbl; 1 oil barrel = 159 L). DWH is the largest accidental marine oil spill in the history of the petroleum industry.2 After DWH, key questions were asked: What is the likelihood that a similar catastrophic oil spill (with a volume over 1 MMbbl) will happen again? Is DWH an extreme event or will it happen frequently in the future? Answers to these questions are essential to oil spill risk assessment, contingency planning, and environmental impact statement preparation on oil exploration, development, and production. To answer these questions, the extreme value theory (EVT) is applied to analyze oil spills in the U.S. outer continental shelf (OCS) in the past 49 years (1964− 2012). The objective is to estimate the likelihood of catastrophic oil spills in OCS areas. Offshore drilling for oil and gas has been conducted since the early 1900s.3 Oil and gas deposits under the seabed continue to be an important part of the energy resources of the United States. The term “outer continental shelf” is a legal term created by federal statute and is distinct from the geographic term “continental shelf”. Legally, the OCS comprises that part of the submerged lands, subsoil, and seabed lying between the seaward extent of the States’ jurisdiction and the seaward extent of federal jurisdiction, which is generally 3 geographic miles seaward from the State’s coastline to about 200−300 nautical miles offshore.4 The majority of OCS activity occurs in the Gulf of Mexico, the Pacific Ocean, and off the coast of Alaska. The western and This article not subject to U.S. Copyright. Published 2014 by the American Chemical Society
central portions of the northern Gulf of Mexico (GOM) make up one of the world’s major oil- and gas-producing areas and have been a steady and reliable source of crude oil and natural gas for more than 50 years.4 The need to balance the value of these resources against the potential for environmental damage is an important concern.5 Understanding and managing risks caused by extreme events is one of the most demanding statistical problems. A common feature of extreme events is that they happen rarely, and few data are available to describe stochastically such events. Therefore, by definition, extreme events do not have sufficient historical data for traditional statistical methods. In the recent years, statistical theory of extreme values, especially the EVT, have been widely used in studying rare events, including damage from hurricanes, floods, sea level rise, stock market crashes, large wildfires, earthquakes, etc. (e.g., refs 6−10). EVT assesses the probability of events that are more extreme than any previously observed (or rarely observed in the past). EVT has made great progress since the work by Fisher and Tippett.11 It provides models relevant for the assessment of rare events, even outside the range of previous observations. Extreme value methodology is now commonly used in a wide range of fields, such as economic damage,12,13 finance,6,7 earth sciences,8,11,12,14,15 traffic prediction,16 large oil spills,17,18 etc. Catastrophic oil spills, which are defined in this study as oil spills of over 1 MMbbl in volume, are rare events with potentially very high costs for society and the environment. A standard way Received: Revised: Accepted: Published: 10505
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to estimate an expectation is by the arithmetic mean (the sum of all observed data divided by the number of data). To do this meaningfully, however, many data are needed. Because catastrophic oil spills are rare, this traditional statistical method cannot be applied simply because of the lack of data. In this study, the lack of data is compensated for using methods from EVT. This paper focuses on how the statistical theory of extreme values can be applied to improve the analysis of oil spill risk. A statistical modeling approach based on the generalized extreme value distributions is advocated for assessing the risk of catastrophic oil spills. EVT is based on the assumption that the OCS spill data used are independently and identically distributed (abbreviated i.i.d.), which requires that no systematical change of oil spill patterns occurs within the observation period. Therefore, to have an accurate and relevant risk analysis, it is essential that i.i.d. oil spill data are used. Factors influencing oil spill may include (1) technology used in oil exploration, production, and transportation, (2) regulations, (3) climatological conditions, (4) response and mitigation measures, and (5) accuracy of oil accidents reported (including bias introduced by underreporting). These factors vary greatly from country to country and from region to region. For instance, among the 347 OCS oil spills with volume over 50 barrels (bbl) in the past 49 years, 32% of them happened in the two months of August and September. This is primarily due to hurricanes, a climatological phenomenon unique in the GOM. In addition, regions/countries with less stringent regulatory requirements on safety and response measures are more likely to have large spills. Therefore, to ensure that the oil spill data are i.i.d., it is essential that the oil spill data from the region(s) with similar characteristics are used. For these reasons, this study focuses on oil spills occurring in OCS areas, including the GOM OCS and the Pacific OCS. Thus far, all oil spills in the Alaska OCS are too small to affect the results of this study. The Exxon Valdez oil spill in Alaska in 1989 did not happen in the OCS area and did not involve oil produced from the OCS; therefore, it was not included in the analysis.19 This study focuses on oil spills that occurred in OCS areas and excludes oil spills in other parts of the world. The primary considerations are as follows: (1) The spills in other parts of the world might not be assumed to follow the same statistical behavior as OCS spills, because the factors influencing oil spills vary greatly. (2) In comparison to other parts of the world, data describing OCS oil spills provide for a longer and much more complete record (1964−2012), which is essential for conducting EVT analysis. Large oil spills are generally rare, discrete events. Although rare, large events account for most of the cumulative oil spill volume, and most oil spills are small. From 1996 to 2010, 96% of OCS spills were less than 1 bbl.16 The DWH volume was more than 8.5 times the cumulative 570 000 bbl that were spilled in the previous 46 years (including all spills more than 1 bbl during the period of 1964−2009) from all OCS oil and gas activities.20 This high contribution of the largest spills to the total spill volume illustrates that mean and accumulated spill volumes do not present the complete picture of the oil spill risk. The objective of this study is to apply EVT to analyzing catastrophic oil spills in OCS areas to determine the likelihood that a similar catastrophic spill could happen again.
Article
MATHEMATICAL METHOD
An oil spill distribution, in practice, may be considered in two possibly distinct parts: (1) the central part of the distribution, which deals with the spills that happen often (such as on an annual basis) (in this case, standard statistical models can be used)21,22 and (2) the extreme end of the oil spill distribution (in this case, models based on EVT are needed). Therefore, a combination of models is needed in oil spill risk analysis. Extreme events can be very different in different areas (such as insurance, stock market, and earth science), but similar tools can be used by practitioners to analyze the risks. The consequences of catastrophic oil spills are highly dependent upon circumstances and may also be viewed differently by different stakeholders. For example, a spill of 10 000 bbl may be considered catastrophic to the operator of a platform (or pipeline or oil tanker), but as that kind of oil spill is typically analyzed by the oil spill risk analysis model (e.g., ref 23), it is not considered extreme in the context of this paper. Spills with a volume over 1 MMbbl are considered as extreme and are the focus of this study. Standard statistical models often use past experience to estimate the consequence of future events. If one is only analyzing the frequency of occurrence of oil spills and not their severity, then the standard statistical techniques can be used.24,25 Empirical distributions do not require any assumptions about the shape of the distribution and, thus, offer a reliable characterization of the risk, particularly if a large set of observations is available. For example, the Poisson distribution is routinely used in calculating the probability of oil spills in OCS areas.21−23 Oil spill risks can then be given as empirical distribution constructed directly from data. However, to factor in the severity of extreme events, the Poisson distribution falls short because prediction of events becomes more difficult when data are incomplete. For example, how can the 1 in 100 year flood be predicted if less than 25 years of data are available? Standard statistical models can suffer in this respect, and results from the models could be misused. Thus, a combination of models is needed in oil spill risk analysis to deal with catastrophic oil spills. The statistical distribution of catastrophic events can only be analyzed by a generalized extreme value (GEV) distribution. This is one of the most fundamental results of EVT. More details are provided by Reiss and Thomas.26 EVT shows that similar tools can be used to analyze the risks of a wide variety of extreme events. Using extreme value methods, it is possible to extrapolate beyond the data for the prediction of events. Unlike standard statistical analysis, where outliers are ignored, such data are precisely what drives extreme events. The key idea is that probabilities of extreme events can be estimated by fitting a “model” to a set of extreme event data, which is just the tail of the complete data distribution (in this paper, the annual oil spill maxima will be used). Then, the model can be used to estimate extreme events. This is a distinguishing aspect for the catastrophic oil spill analysis, where very few observations are available. This approach can also estimate the frequency and volume of potential future catastrophic oil spills that might exceed the historical worst cases. Specific methods based on the EVT are needed for the statistical treatment of rare events, such as the tail behavior, and to compensate for the insufficient data available. Two methods are often used in the statistics of extremes. The first is called the blocks method, and the second is the peaks-over-thresholds (POT) method. The blocks method relies on deriving block 10506
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the maximum-likelihood (ML) method26 in fitting the annual maxima of oil spills.
maxima series as a preliminary step. It is often convenient to extract the annual maxima, as is the case in the present study. The POT method involves fitting two distributions: one for the number of events in a basic time period and the second for the volume of the exceedances. The preferred method depends upon the question posed and the data at hand. For example, Eckle et al.18 used the POT method to analyze 37 years of data (from 1974 to 2010) on large oil spills globally. In comparison to the modeling of annual extremes (the blocks method), the POT method has positive and negative properties. On the one hand, taking all exceedances of a sample usually gives more observations. On the other hand, such exceedances can occur in clusters, so that the i.i.d. property could be violated. For example, a total of 347 spills with volume over 50 bbl happened in OCS areas in the past 49 years (1964−2012). A total of 25 of the 347 spills happened on a single day (August 29, 2005) because of Hurricane Katrina. Obviously, these 25 spills are all related and are not i.i.d. More detailed consideration is necessary before including these 25 spills into such an analysis. To avoid the effects of dependence, declustering procedures (i.e., making use only of the single highest exceedance within a cluster) may be employed in applications of the POT approach. The POT method may also lead to some years without data. For these reasons, the blocks method is used in this study by assuming that annual maxima of oil spills in OCS areas are i.i.d. In this way, every year is a block and the data used are block maxima (i.e., the single highest oil spill volume over an entire year). To understand the blocks method, four terms need to be defined: probability density function (PDF), cumulative distribution function (CDF), quantile function, and the GEV family of functions. A PDF of a continuous random variable is a function that describes the relative likelihood for this random variable to have a given value. The PDF is non-negative, and its integral over the entire space is equal to 1. A CDF describes the probability that a random variable will be found at a value less than or equal to a certain value (x). In the case of a continuous distribution, it gives the area under the PDF from −∞ to x. A quantile function is another way of prescribing a probability distribution. For a given PDF value (probability) of the random variable, the quantile function specifies the value at which the random variable will be less than or equal to that probability. The quantile function is the inverse of its CDF and is also called the percent point function or inverse cumulative distribution function. The GEV family of functions describes the distribution of the block maximum of a set of data, which are a series of i.i.d. observations. The CDF of the GEV is given by
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RESULTS Figure 1 shows the time series of annual largest oil spills derived from OCS data for 49 years from 1964 to 2012. Because of the
Figure 1. Time series of annual largest oil spills derived from OCS data for 49 years from 1964−2012. The dotted line is the 10 year return level, and the dashed line is the 1 year return level.
large variation of oil spill volume, the annual maximum oil spill is represented on a log scale. The dotted line is the 10 year return level, and the dashed line indicates the 1 year return level, which will be discussed later in this paper. A log scale is commonly used to represent large variations in EVT studies. For example, McNeil12 used a log scale to present Danish fire loss data and to conduct the extreme value analysis; Fasen et al.7 applied a log scale to present the negative profit/loss of the market portfolio and introduced a variable of the daily negative log returns; and Eckle et al.18 used a log scale to describe oil spill volume and global exceedance frequency. The GEV distribution of eq 1 is fitted by the ML method to the log transformation of the annual maxima. The R software is used to carry out the statistical analysis without any manual adjustment to the parameters. The location parameter (μ) is 2.442 (which is equivalent to 277 bbl); the scale parameter (σ) is 0.839; and the shape parameter (γ) is −0.073. To assess the model fit graphically, a probability−probability plot (PP plot) (Figure 2) and a quantile−quantile plot (QQ plot) (Figure 3) are used for the transformed data set. Figure 2 presents the probability (or percentile) of the empirical distribution for each quantile value against the modeled probability of the GEV for the same value. The line of equality indicates a perfect fit. The closeness of the PP plot to this line is a measure of goodness of fit. As an example, the value of 1456 bbl is the 50th percentile of the empirical distribution and the 48th percentile (approximately) of the GEV. Figure 3 presents the distribution fit by a QQ plot for the estimated GEV parameters. In Figure 3, the quantiles of the modeled GEV distribution are plotted against the empirical quantiles based on the historical data. Again, the line of equality
⎧ exp{−[1 + γ(x − μ)/σ ]−1/ γ }, ⎪ F(x ; μ , σ , γ ) = ⎨ 1 + γ(x − μ)/σ > 0, γ ≠ 0 ⎪ ⎩ exp{−exp[− (x − μ)/σ ]}, γ = 0 (1)
The GEV distribution has three parameters: (1) the location parameter (μ) specifies the center of the distribution; (2) the scale parameter (σ) determines the size of deviations about the location parameter; and (3) the shape parameter (γ) governs how rapidly the upper tail decays. In this study, eq 1 is used to estimate the largest annual oil spill in OCS areas. The behavior of the annual maximum is then approximately described by one of the GEV family of distributions given by the CDF F(x; μ, σ, γ). The statistical software language and environment called R27,28 was used to estimate location, scale, and shape parameters using 10507
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Figure 4. Return levels under different return periods. The dots are calculated on the basis of the observations, and the curve is from the GEV distribution calculated using the ML method with the R software.
Figure 2. PP plot for fit of the GEV distribution to annual oil spill maxima (line of equality indicates a perfect fit).
levels for the 10 and 100 year return periods are 16 and 562 Mbbl (1000 bbl), respectively. The return level of 1 MMbbl (i.e., the catastrophic oil spill) corresponds to a return period of 165 years. Therefore, on the basis of the past 49 years of OCS oil spill data, the GEV distribution indicates that a catastrophic oil spill in OCS areas is most likely to occur with a return period of 165 years. Figure 1 compares the 10 year return level (dotted line) with the 49 annual maxima. There are five annual maxima higher than the 10 year return level, about one in every 10 years on average. This is consistent with the 10 year return level of 16 Mbbl estimated from the GEV distribution. The 1 year return level of 277 bbl is also shown in Figure 1. To study the sensitivity of the modeled GEV distribution to parameters and observational data, three cases are investigated and compared: (1) DWH case (this is the benchmark case with 49 years of oil spill records from 1964 to 2012, including the DWH spill; the results are already presented in Figures 2−4), (2) no DWH case (this is the case of assuming that DWH did not happen in 2010, and the second largest spill in 2010, which is 123 bbl, is used in the analysis), and (3) 10 MMbbl case (this is the case of assuming that a spill of 10 MMbbl happened in 2013, and there are a total of 50 annual maxima from 1964 to 2013). Figure 5 is the same as Figure 2, except that the cases of no DWH and 10 MMbbl are added. It is evident that, for all three cases, modeled probabilities from the GEV distributions fit the empirical probabilities based on the observational data reasonably. Figure 6 is the same as Figure 3, except that the cases of no DWH and 10 MMbbl are added. The modeled quantiles also match the empirical quantiles well. Therefore, Figures 5 and 6 show that the GEV distributions obtained from the data sets are capable of representing the oil spill distributions reasonably well. Figure 7 is the same as Figure 4, except that the cases of no DWH and 10 MMbbl are added. The three cases have similar return levels with shorter return periods, when the return period is less than 10 years or so. This result is expected, because the differences between the cases are in the spills with a volume over 1 MMbbl. When the return period is 20 years or longer, the
Figure 3. QQ plot for fit of the GEV distribution to annual oil spill maxima (line of equality indicates a perfect fit).
indicates perfect fit. There are 49 annual maxima in the OCS oil spill data set. Again, the value of 1456 bbl is equivalent to the median (50th percentile) in the empirical distribution. This value is plotted against the 50th percentile of the modeled GEV distribution, which is 1374 bbl. Both Figures 2 and 3 indicate that the fits are good and follow the line of equality reasonably. The return level (or return value) of a random variable is the quantile value, which is exceeded, on average, once in a period of time (called the return period). For example, the return period (such as 100 year flood) based on extreme precipitation (i.e., certain return value) is commonly used to assess the capacity of drainage systems. Figure 4 gives return levels under different return periods. Figure 4 (and Table 2 later) show that the return 10508
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Figure 7. Same as Figure 4, except that the cases of no DWH and 10 MMbbl are added. Figure 5. Same as Figure 2, except that the cases of no DWH and 10 MMbbl are added.
Table 1. Return Periods of Catastrophic Oil Spills in OCS Areas case
return period (years)
confidence interval (years) (5−95%)
DWH no DWH 10 MMbbl
165 >500 86
from 41 to >500 >500 from 20 to >500
of 10 MMbbl, the return period is 86 years, with a 95% confidence interval between 20 years and more than 500 years. For model sensitivity analysis, Table 2 summarizes the return levels with return periods of 10 and 100 years in the three Table 2. Return Levels with Return Periods of 10 and 100 Years in OCS Areasa return level (Mbbl) (5−95%) return period (years)
DWH
no DWH
10 MMbbl
10 100
16 (3−89) 562 (16−11220)
15 (6.9−53) 122 (27−815)
13 (2.4−43) 1301 (35−372759)
a
The values in parentheses are the 95% confidence intervals.
different cases (DHW, no DWH, and 10 MMbbl). Table 2 shows that with a return period of 10 years, the three cases all have similar return levels (in Mbbl), varying from 16 for the DWH case to 13 for the 10 MMbbl case. This is another indication that the GEV distributions obtained from the OCS data are relatively stable. The 95% confidence intervals (in Mbbl) are 3−89, 6.9− 53, and 2.4−43, respectively. For the return period of 100 years, the return levels (in Mbbl) are 562, 122, and 1301 for DHW, no DHW, and 10 MMbbl cases, respectively.
Figure 6. Same as Figure 3, except that the cases of no DWH and 10 MMbbl are added.
return levels can be very different, because the large spills change the long period part of the GEV distribution. Table 1 summarizes the return periods (and the 95% confidence intervals) of the catastrophic spill in the three cases. It reveals that operations in OCS areas are most likely to result in a catastrophic oil spill (with the volume of 1 MMbbl or more) once every 165 years. The 95% confidence interval is between 41 years and more than 500 years. Because there are 49 years of observational data, it is not that meaningful to estimate periods longer than 500 years. In the case of no DWH, the return period is expected to be more than 500 years. The lower bound of the 95% confidence interval is also more than 500 years. In the case
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DISCUSSION This paper intended to answer these key questions: (1) What is the likelihood that a catastrophic oil spill similar to DWH will happen again? (2) Is DWH an extreme event or will it happen frequently in the future? Answers to these questions are essential to oil spill risk assessment, contingency planning, and environmental impact statement preparation related to oil exploration, 10509
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(10) McNeil, A. J.; Frey, R.; Embrechts, P. Quantitative Risk Management: Concepts, Techniques and Tools; Princeton University Press: Princeton, NJ, 2005. (11) Fisher, R. A.; Tippett, L. H. C. Limiting forms of the frequency distribution of the largest or smallest member of a sample. Proc. Cambridge Philos. Soc. 1928, 24, 180−190. (12) McNeil, A. Estimating the tails of loss severity distributions using extreme value theory. ASTIN Bull. 1997, 27, 117−137, DOI: 10.2143/ AST.27.1.563210. (13) Jagger, T. H.; Elsner, J. B.; Saunders, M. A. Forecasting US insured hurricane losses. In Climate Extremes and Society; Diaz, H. F., Murnane, R. J., Eds.; Cambridge University Press: Cambridge, U.K., 2008; pp 189−208. (14) Dorland, C.; Tol, R. S. J.; Palutikof, J. P. Vulnerability of the Netherlands and Northwest Europe to storm damage under climate change. Clim. Change 1999, 43, 513−535. (15) Katz, R. W. Statistics of extremes in climate change. Clim. Change 2010, 100, 71−76, DOI: 10.1007/s10584-010-9834-5. (16) Zheng, L.; Ismail, K.; Meng, X. Freeway safety estimation using extreme value theory approaches: A comparative study. Accid. Anal. Prev. 2014, 62, 32−41, DOI: 10.1016/j.aap.2013.09.006. (17) Stewart, R. J.; Kennedy, M. B. An Analysis of U.S. Tanker and Offshore Petroleum Production Oil Spillage through 1975; Martingale, Inc.: Cambridge, MA, 1978. (18) Eckle, P.; Burgheer, P.; Michaux, E. Risk of large oil spills: A statistical analysis in the aftermath of Deepwater Horizon. Environ. Sci. Technol. 2012, 46, 13002−13008. (19) Anderson, C. M.; LaBelle, R. P. Comparative occurrence rates for offshore oil spills. Spill Sci. Technol. Bull. 1994, 1, 131−141. (20) Anderson, C. M.; Mayes, M.; LaBelle, R. P. Update of Occurrence Rates for Offshore Oil Spills; United States Department of Interior (USDOI): Herndon, VA, 2012; OCS Report BOEM 2012-0069, http://www.boem.gov/uploadedFiles/BOEM/Environmental_ Stewardship/Environmental_Assessment/Oil_Spill_Modeling/ AndersonMayesLabelle2012.pdf. (21) Smith, R. A.; Slack, J. R.; Wyant, T.; Lanfear, K. J. The Oil Spill Risk Analysis Model of the U.S. Geological Survey; United States Government Printing Office: Washington, D.C., 1982; Geological Survey Professional Paper 1227, http://www.boem.gov/EnvironmentalStewardship/Environmental-Assessment/Oil-Spill-Modeling/ smithetal-pdf.aspx. (22) Ji, Z.-G.; Johnson, W. R.; Marshall, C. F.; Rainey, G. B.; Lear, E. M. Oil-Spill Risk Analysis: Gulf of Mexico Outer Continental Shelf (OCS) Lease Sales, Central Planning Area and Western Planning Area, 2003− 2007, and Gulfwide OCS Program, 2003−2042; Minerals Management Service: Herndon, VA, 2002; OCS Report 2002-032. (23) Ji, Z.-G.; Johnson, W. R.; Li, Z. Oil spill risk analysis model and its application to the Deepwater Horizon oil spill using historical current and wind data. In Monitoring and Modeling the Deepwater Horizon Oil Spill: A Record-Breaking Enterprise; Liu, Y., Macfadyen, A., Ji, Z.-G., Weisberg, R. H., Eds.; American Geophysical Union: Washington, D.C., 2013; DOI: 10.1029/2011GM001117. (24) Dekking, F. M.; Kraaikamp, C.; Lopuhaä, H. P.; Meester, L. E. A Modern Introduction to Probability and Statistics; Springer-Verlag: London, U.K., 2005. (25) Ji, Z.-G. Hydrodynamics and Water Quality: Modeling Rivers, Lakes, and Estuaries; John Wiley and Sons, Inc.: Hoboken, NJ, 2008. (26) Reiss, R.-D.; Thomas, M. Statistical Analysis of Extreme Values with Applications to Insurance, Finance, Hydrology and Other Fields, 2nd ed.; Birkhauser: Basel, Switzerland, 2001. (27) R Development Core Team. R: A Language and Environment for Statistical Computing; R Foundation for Statistical Computing: Vienna, Austria, 2013; ISBN: 3-900051-07-0. (28) Gilleland, E.; Ribatet, M.; Stephenson, A. G. A software review for extreme value analysis. Extreme 2013, 16, 103−119, DOI: 10.1007/ s10687-012-0155-0.
development, and production. In the past 10 years, a statistical theory of extreme values has been widely used in extreme event analyses in a variety of areas, including science, engineering, finance, and insurance. In this paper, the generalized extreme value distributions are applied to analyze oil spills in the OCS in the past 49 years (1964−2012). Major findings from this study include the following: (1) The blocks method from the EVT is very useful for oil spill risk analysis. To the knowledge of the authors, this is the first published study in which the blocks method is used in analyzing catastrophic oil spills. On the basis of the 49 years of OCS oil spill data, the GEV distributions are capable of describing oil spills in OCS areas. (2) The return period of a catastrophic oil spill in OCS areas is estimated to be 165 years, with a 95% confidence interval between 41 years and more than 500 years. (3) The model sensitivity studies indicate that the GEV distributions obtained from the OCS data are relatively stable. The three cases, DWH, no DWH, and 10 MMbbl, all have similar return levels when the return periods are 10 years or less. When the return periods are 20 years or longer, the three cases yield different return levels. For a return period of 100 years, the return levels (in Mbbl) of these three cases are 562, 122, and 1301, respectively. The statistical modeling of extremes remains a subject of active research. From the perspective of oil spill risk analysis, much remains to be done. Oil spill risks vary from region to region and from operation to operation. Future studies on catastrophic oil spills should consider oil production, seasonality, and spatial dependence. Methods considering multiple influencing factors should be used to include these effects.
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AUTHOR INFORMATION
Corresponding Author
*Telephone: 703-787-1145. Fax: 703-787-1053. E-mail: jeff.ji@ boem.gov. Notes
The authors declare no competing financial interest.
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REFERENCES
(1) United States Coast Guard. On Scene Coordinator Report: Deepwater Horizon Oil Spill; United States Coast Guard: Washington, D.C., 2011; http://www.uscg.mil/foia/docs/dwh/fosc_dwh_report. pdf. (2) National Commission on the BP Deepwater Horizon Oil Spill and Offshore Drilling. Deep Water: The Gulf Oil Disaster and the Future of Offshore Drilling; National Commission on the BP Deepwater Horizon Oil Spill and Offshore Drilling: Washington, D.C., 2011; www. oilspillcommission.gov. (3) Prutzman, P. W. Petroleum in Southern California; California State Mining Bureau: Sacramento, CA, 1913. (4) LaBelle, R. P. Overview of US Minerals Management Service activities in deepwater research. Mar. Pollut. Bull. 2001, 43, 256−261. (5) Ji, Z.-G. Use of physical sciences in support of environmental management. Environ. Manage. 2004, 34 (2), 159−169. (6) Sanders, D. E. A. The modelling of extreme events. Br. Actuarial J. 2005, 519−557, DOI: 10.1017/S1357321700003251. (7) Fasen, V.; Klüppelberg, C.; Menzel, A. Quantifying extreme risks. In Risk: A Multidisciplinary Introduction; Klüppelberg, C., Straub, D., Welpe, I., Eds.; Springer: Dordrecht, Netherlands, 2014; pp 151−181. (8) Katz, R. W.; Parlange, M. B.; Naveau, P. Statistics of extremes in hydrology. Adv. Water Resour. 2002, 25, 1287−1304. (9) Pickands, J., III Statistical inference using extreme order statistics. Ann. Stat. 1975, 3 (1), 119−131. 10510
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