rT"ii y^\*i *m *i~ii * The OilsDill Ri

of the U.S. Geological " -Gr Survey/

GEOLOGICAL StlltVEY PROFESSIONAL FAJFE
The Oilspill Risk Analysis Model 

of the U.S. Geological Survey

By RICHARD A. SMITH, JAMES R. SLACK, TIMOTHY WYANT, and 

KENNETH J. LANFEAR

GEOLOGICAL SURVEY PROFESSIONAL PAPER 1227

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON: 1982
UNITED STATES DEPARTMENT OF THE INTERIOR 

JAMES G. WATT, Secretary

GEOLOGICAL SURVEY 

Dallas L. Peck, Director

Library of Congress Cataloging in Publication Data

Main entry under title:

The Oilspill risk analysis model of the U.S. Geological Survey.

(Geological Survey Professional Paper 1227)

Bibliography: p. 35

Supt. of Docs, no.: I 19.16:1227

I. Oil spills Mathematical models. 2. Oil spills Data processing.

1. Smith, Richard A. II. Series.

GCI085.044 363.7'394 80-606812 AACR2

For sale by the Branch of Distribution, U.S. Geological Survey, 

604 South Pickett Street, Alexandria, VA 22304
CONTENTS

Abstract .................................................... 1

Introduction ................................................. 1

Representations of physical data .... ........................... 2

Base map ............................................... 2

Land and targets ......................................... 3

Storage of targets ..................................... 3

Types of targets ...................................... 3

Further refinements of targets seasonal vulnerability ...... 3

Land segments ....................................... 4

Checking target and land segment data ................... 5

Insertion of spatial data into the model ................... 7

Paging system for large arrays .......................... 8

Winds .................................................. 8

Stochastic model of wind data ........................... 8

Constructing wind transition matrices: programs

RAWWIND, LISTWIND, and WINDTRAN .......... 10

Defining wind zones: program WINDZONE .............. 10

Currents ................................................ 11

Current data entry and storage: programs CURPOLY and

CURMATRX ..................................... 11

Current data checking ................................. 11

Oilspill trajectory simulation ................................... 11

Monte Carlo simulations of oilspill trajectories ................. 11

Effects of wind and current on oilspill movement ............... 13

Oilspill movement ........................................ 15

Risk calculation .............................................. 17

Conditional probabilities................................... 17

Conditional probabilities of contacting targets:

program HITPROB ................................ 17

Conditional probabilities of contacting land segments:

program LANDSEG ................................ 18

Travel times for oilspills contacting targets:

Program FIRSTPAS ................................ 18

Risk calculation Continued

Spill occurrence .................................

Some basic features of spill occurrence forecasting .

Predicted probability distribution for a fixed class

of spills ..................................

Choosing an exposure variable .....................

Spill occurrence rates and exposure variables .........

Transportation Scenarios .............................

Construction of transportation scenarios:

program SCENARIO ..........................

Estimated volumes of oil reserves ...................

Probability that an oilspill will occur and contact a target . . .

Probability of hits on a target from a single source .....

Probability of hits on a target from multiple spill sources 

Model verification and limitations ..........................

Formal error analysis.................................

Informal error analysis ...............................

Spatial resolution ................................

Risk from near-shore and confined-area spill sources . . .

Spreading 

Decay . . .

Sensitivity analysis ..........................

Direct model verification .....................

Model output and case examples ..................

Reports for OCS leasing .....................

Summary of results to date ...................

Other possible uses of the model ...............

Practical aspects of operating and managing the model

Management system ........................

Software ..................................

Hardware .................................

Selected references ..............................

Appendix Distribution theory of spill incidence .....

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ILLUSTRATIONS

Page

FIGURE 1-3. Maps showing:

1. Sample target in the Western Gulf of Alaska .......................................................................... 5

2. Sample target in the Eastern Gulf of Mexico .......................................................................... 6

3. Typical division of the shoreline into land segments .................................................................... 7

4. Example of a wind transition matrix for winter at Pt. Arguello, Calif ......................................................... 9

5-8. Maps showing:

5. Wind zones used for OCS Lease Sale 48 southern California ........................................................... 12

6. Monthly current field for southern California (March).................................................................. 13

7. Current polygons for southern California, developed from the monthly current field ......................................... 14

8. Example oilspill trajectories for a spill site near the center of the proposed Mid-Atlantic (OCS Lease Sale 49) lease area: summer conditions ........................................................................................................ 16

9. Movement of a hypothetical oilspill through the model's grid system during one time step ........................................ 18

10. Map showing example oilspill trajectories for a spill site near southern California (OCS Lease Sale 48) ............................. 19

11. Map illustrating potential transportation route segments for southern California, showing how oil from tract group P14 would be brought

to Long Beach via segments T20 and T21 ........................................................................... 25

12. Comparison of the observed slick from the Argo Merchant with a prediction of the Oilspill Risk Analysis Model ..................... 30

13. Flow chart illustrating the major elements of a complete model run .......................................................... 34
IV CONTENTS

TABLES

Page

TABLE 1. Targets for a risk analysis in the Western Gulf of Alaska ..................................................................... 4

2. Targets for a risk analysis in the Eastern Gulf of Mexico ..................................................................... 4

3-5. Examples of:

3. Typical output from program HITPROB, showing probabilities that an oilspill starting at a particular location will reach a certain 

target in 30 days .................................................................................................. 20

4. Typical output from program LANDSEG, showing probabilities that an oilspill starting at a particular locatipn will reach a certain land 

segment in 30 days ................................................................................................ 21

5. Output from program FIRSTPAS. Average minimum, maximum, and mean times-of-travel for oilspills occurring at site PI to contact 

targets .......................................................................................................... 22

6. Historic spill occurrence rates used in the Oilspill Risk Analysis Model ......................................................... 23

7. Sensitivity of predicted oilspill risks for the North Atlantic study area to the assumption that winds can be modeled as a first-order Markov

process.......................................................................................................... 29

8. List of reports prepared for OCS lease sale analyses using the Oilspill Risk Analysis Model ......................................... 29

9. Estimated conditional probabilities and expected number of spills larger than 1,000 barrels reaching shore as a result of petroleum development in each of the six Federal lease areas ............................................................................. 31

10. Expected number of oilspill contacts with and impacts on coastal and marine resources in the six Federal lease areas .................... 33

METRIC CONVERSION FACTORS

SI (International System of Units) is a modernized metric system of measurement. An asterisk after the last digit of the factor 

indicates that the conversion factor is exact and that all subsequent digits are zero, all other conversion factors have been 

rounded to four significant digits.

To convert from____________________________To______________________Multiply by_________

inch (in) .................... millimeter (mm) ............. 25.4*

foot (ft) ................... .meter (m)................... 0.3048

mile (mi) .................. . kilometer (km) .............. 1.609

nautical mile ................ kilometer (km) .............. 1.852*

acre ....................... meter2 (m2 ) ................. 4,047

mile2 (m2 ) .................. .kilometer2 (km2 ) ............. 2.590

gallon (gal) ................ .meter3 (m3 ) ................. 0.003785

barrel(bbl) ................ .meter3 (m3 ) ................. 0.1590

(petroleum,

1 bbl = 42 gal) 

million gallons ............. . meter3 per second (m3/s) ....... 0.04381

per day (Mgal/d)
THE OILSPILL RISK ANALYSIS MODEL 

OF THE U.S. GEOLOGICAL SURVEY

By RICHARD A. SMITH, JAMES R. SLACK, 

TIMOTHY WYANT, and KENNETH J. LANFEAR

ABSTRACT

The U.S. Geological Survey has developed an oilspill risk analysis 

model to aid in estimating the environmental hazards of developing 

oil resources in Outer Continental Shelf (OCS) lease areas. The large, 

computerized model analyzes the probability of spill occurrence, as 

well as the likely paths or trajectories of spills in relation to the locations of recreational and biological resources which may be vulnerable. The analytical methodology can easily incorporate estimates 

of weathering rates, slick dispersion, and possible mitigating effects 

of cleanup.

The probability of spill occurrence is estimated from information 

on the anticipated level of oil production and method and route of 

transport. Spill movement is modeled in Monte Carlo fashion with a 

sample of 500 spills per season, each transported by monthly surface-current vectors and wind velocities sampled from 3-hour windtransition matrices. Transition matrices are based on historic wind 

records grouped in 41 wind velocity classes, and are constructed 

seasonally for up to six wind stations. Locations and monthly vulnerabilities of up to 31 categories of environmental resources are digitized within an 800,000 km2 study area. Model output includes 

tables of conditional impact probabilities (that is, the probability of 

hitting a resource, given that a spill has occurred), as well as probability distributions for oilspills occurring and contacting environmental resources within preselected vulnerability time horizons.

The model provides the U.S. Department of the Interior with a 

method for realistically assessing oilspill risks associated with OCS 

development. To date, it has been used in oilspill risk assessments 

for eight OCS lease sales with the results reported in Federal environmental impact statements. A summary of results is presented 

herein. A "real time" version was also used to forecast the movement of oil from the 1976-77 Argo Merchant oilspill. Additional 

model runs are planned for future OCS lease sales in frontier areas. 

Other possible applications include analysis of OCS development alternatives and site selection for oilspill cleanup equipment.

INTRODUCTION

The past decade has been a period of rapid growth in 

the offshore petroleum industry. The Department of 

the Interior currently conducts sales of mineral leases

for specific areas of the Outer Continental Shelf at the 

rate of more than two per year, and it is anticipated 

that lease sales will continue, perhaps even at an increased rate, well into the 1980's.

Oilspills are one of the major concerns associated 

with offshore oil development in all OCS lease sale 

areas. Concern is clearly strongest among those who 

live in coastal areas and who depend, directly or indirectly, on coastal zone resources other than oil for a 

livelihood. Controversy over the risks and benefits of 

off-shore oil development inevitably gives rise to a 

need for quantitative estimates of the oilspill risk involved in a particular development proposal. Within 

the Federal Government, oilspill risk estimates are required prior to holding an OCS lease sale, at the time 

the Secretary of the Interior makes decisions on tracts 

to be withheld from leasing because of unacceptable 

oilspill risk to specific environmental resources in the 

proposed sale area. At issue in the decisionmaking for 

a typical OCS lease sale are anywhere from 100 to 500 

nine-square-mile tracts which have been identified as 

possible production areas by interested oil companies. 

Also at issue are as many as 20 or 30 specific resources 

which have been identified by the Bureau of Land 

Management or the U.S. Geological Survey as vulnerable to oilspills on the basis of research and communication with local authorities.

An important fact that stands out when one attempts to predict oilspill damages for a proposed OCS 

lease area is that the problem is fundamentally probabilistic. A great deal of uncertainty exists not only 

with regard to the location, number, and size of spills 

that will occur during the course of development, but 

also with regard to the wind and current conditions 

that will exist and give direction to the oil at the particular times spills occur. While some of the uncertainty
OILSPILL RISK ANALYSIS MODEL OF THE U.S. GEOLOGICAL SURVEY

reflects incomplete or imperfect data, considerable uncertainty is simply inherent in the problem.

The Geological Survey has developed a model for assessing the oilspill risks associated with petroleum development in Federal OCS lease areas. The model is 

constructed to deal with three fundamental and essentially independent factors which comprise the total oilspill risk to coastal zone resources: (1) the probability 

of spill occurrence as a function of the quantity of oil 

which is to be produced and handled at individual production sites, pipelines, and tanker routes; (2) the probabilities of occurrence of various spill trajectories from 

production sites and transportation routes as a function of historical wind and current patterns for the 

area; and (3) the location in space and time of vulnerable resources defined according to the same coordinate system used in spill trajectory simulation. Results 

of the individual parts of the analysis are combined to 

estimate the total oilspill risk associated with production and transportation at locations within a proposed 

lease area. This information is then used in making final tract selections prior to leasing. To date, risk analyses have been conducted for seven Federal lease areas, 

including sites offshore the North-, Mid-, and SouthAtlantic Coasts, the Eastern Gulf of Mexico, Southern 

California, and the Western and Northern Gulf of Alaska.

The purpose of this report is to describe how the Oilspill Risk Analysis Model of the U.S. Geological Survey works, both in theory and in actual operation. It 

discusses the assumptions used in developing the model and defines the role of each computer program. 

While not a detailed operating instruction manual, it 

provides the broad understanding of the model which 

is necessary for operating the model and properly interpreting the results.

The report begins with a discussion of how the data 

base is developed, proceeds to describe how oilspills 

are simulated, and then reviews the results to date. 

The section, "Representations of Physical Data," describes how winds, currents, and the locations of environmental resources, or targets, are represented as data and put in the proper form for analysis. Simulation 

of oilspill movement is the topic of the section, "Oilspill Trajectory Simulation," and the probabilistic calculations of oilspill risk is covered in "Risk Calculation." The section, "Model Verification and Limitations," places the accuracy of risk calculations in perspective with discussions of sensitivity and verification studies. A summary of past results and ideas for 

future uses of the model are presented in the section, 

"Model Output and Case Examples." Discussion of 

"Practical Aspects of Operating and Managing the 

Model" concludes the paper.

REPRESENTATIONS OF PHYSICAL DATA

The model of the U.S. Geological Survey is designed 

to use a large amount of information about the physical environment, including sizable files of wind and 

current data and the locations of numerous environmental resources which may be adversely affected by 

oilspills. Model programs process all of this data and 

store it in computer files before any trajectories are 

computed. All of the files are designed to allow rapid 

access to the data by subsequent computer programs. 

An extensive system of internal checks, along with 

graphic displays and printouts, help ensure that physical data are represented correctly. The following section describes how physical data are collected, processed, checked, and stored.

BASE MAP

A system for representing spatial locations is the 

foundation of the trajectory simulation model. The model employs a Cartesian coordinate system superimposed over a base map of the study area. All stored data are referenced to this system, and it is used for all internal calculations.

The initial step in establishing a coordinate system 

is the delineation of the area to be modeled. This area 

must be large enough so that all oilspill targets likely 

to be affected, such as land or biological resources, are 

included; at the same time, the map scale must not be 

so large that essential details are obscured. Previous 

OCS lease sale analyses have typically examined areas 

of about 800 km by 800 km, and included 1,000 km of 

coastline. The base map boundaries are usually chosen 

so that the major origins of potential spills, such as the 

lease area and transportation routes, are centered; if 

winds or currents are expected to drive spills predominantly in a certain direction, the map is shifted accordingly. Land need only be included to the extent necessary to define the shoreline, and to aid in visual recognition of the map.

Choice of a projection for the base map is particularly important, since representing the surface of the 

earth by a planar surface necessarily introduces some 

distortion in scale, or direction, or both. The Universal 

Transverse Mercator (UTM) projection system has relatively little scale distortion but has a directional distortion of about 10 degrees. Because the equations for 

correcting this distortion are lengthy and too expensive to perform for each trajectory movement, earlier 

OCS lease sale analyses used UTM or Lambert projections and neglected distortion. However, neglecting 

distortion caused serious difficulties in combining data 

obtained from different maps, and necessitated use of 

a more general mapping system.
REPRESENTATIONS OF PHYSICAL DATA

A useful property of the Mercator projection is that 

there is no distortion in direction; that is, a constant 

compass direction is a straight line. This makes it extremely easy to align a Mercator projection with a Cartesian coordinate system. The penalty for this, however, is extreme distortion in scale, particularly at high 

latitudes. Fortunately, the correction factor is a relatively simple function of latitude, which the computer 

can calculate quickly and easily. Because of these 

properties, the Mercator projection is ideal for oilspill 

modeling purposes, and is now used by the model 

whenever possible.

Once the base map has been selected, a Cartesian coordinate system is superimposed with its origin at the 

lower left-hand (southwest) corner of the map. The 

longest side of the map is usually assigned a length of 

480 units. The whole study area is then divided into a 

matrix of square cells of one unit each; the maximum 

size of this matrix is 480X480 cells. For a typical analysis, each cell represents an area of approximately 2 

to 4 km2 , which is thus the basic unit of resolution for 

spatial data.

Spatial data is stored in a set of 480X480 matrices. 

Elements of the matrices define, for every cell:

 Presence or absence of land, and land segments.

 Presence or absence of up to 31 targets.

 Identification of a wind station, for determining 

the appropriate wind vector for oilspill movement (see subsection Wind Data).

 Identification of a current polygon, for determining the appropriate current vector for oilspill 

movement (see subsection Current Data 

Checking).

Processing data to construct large arrays is a complicated task requiring a great deal of automation. Likewise, the practical limitations of computers require an 

efficient, though sometimes complex, storage and paging scheme for handling these matrices. Other sections 

describe the matrices in more detail.

LAND AND TARGETS

One primary function of the model is to relate oilspill 

trajectory movements to the locations of wildlife populations, fishing areas, and other potential "targets" in 

coastal and continental shelf areas. Environmental impact statements for Federal OCS leasing require collecting an enormous quantity of data about these resources, and a substantial part of this data base 

becomes input for the model.

STORAGE OF TARGETS

The model stores indicators of the presence or 

absence of land and up to 31 other targets in each of a

quarter million grid cells. This is done in such a way 

that each of perhaps 150,000 simulated spills are 

quickly checked at each step in the trajectory for possible impact on each target.

Two features of the model allow a high level of performance in checking cells. When trajectories are being 

simulated by program SPILL (see section on "Oilspill 

Movement,") a paging system burdens computer 

memory with only a small, easily accessible fraction of 

the total grid at any time. Additionally, an effective exploitation of IBM storage attributes provides a compact and efficient mechanism for handling data which 

resides either in main memory or on permanent storage 

devices.

More technically, each grid cell is assigned one 

4-byte integer to indicate the presence of up to 31 

categories of targets, and land. Each of the 32 bits 

(numbered 0-31) corresponds to a different target, or 

land. Bit 0, the sign bit, corresponds to land, and is 

"on" when land is present in the cell. Bit i represents 

the target number i and the interger value 2**(31-i); 

"on" signals that the target is present in the cell. Thus 

an integer value of, say, 9 (binary 00000000 00000000 

00000000 00001001) would indicate that targets 28 

and 31 are present. Simple subroutines can decode 

these integers to suit various purposes.

TYPES OF TARGETS

Examples of spill-vulnerable targets which have been 

included in past analyses appear in tables 1 and 2. 

Sample targets are shown in figures 1 and 2. A simulated spill registers either "hit" or "no hit" on a target. 

A hit is scored as soon as the simulated spill crosses a 

cell occupied by the target. Multiple crossings by the 

same spill count as a single hit.

The selection of targets is clearly of critical importance if the model is to produce useful results. The section, "Model Output and Case Examples" further discusses the targets considered in past risk analyses.

FURTHER REFINEMENTS OF TARGETSSEASONAL VULNERABILITY

Passage of spilled oil through a target location does 

not necessarily imply an adverse impact on the target, 

since vulnerability of a single target may vary according to time of year. Many wildlife populations undergo migrations during the year, and seasonal reproductive activities are often more susceptible to damage 

from spilled oil than other parts of the life cycle. The 

economic impact of spilled oil on such targets as 

beaches may also differ seasonally.
OILSPILL RISK ANALYSIS MODEL OF THE U.S. GEOLOGICAL SURVEY

TABLE 1. Targets fora risk analysis in the Western Gulf of Alaska 

(from Slack, Smith, and Wyant, 1977)

Salmon purse seining and set net areas

Pink and chum salmon intertidal spawning areas

Dungeness crab spawning, rearing, and catch areas

Tanner crab fishing areas

Tanner crab mating and hatching areas

Tanner crab vital rearing areas

Tanner crab important rearing areas

King crab mating and hatching areas

King crab vital rearing areas

King crab important rearing areas

Shrimp fishing areas

Shrimp production rearing areas

Seabird colonies

Summer bird distribution (June, July, August)

Fall bird distribution (September, October, November)

Winter bird distribution (December, January, February)

Spring bird distribution (March, April, May)

Marine mammal foraging areas

Sea lion rookeries and hauling grounds

Harbor seal rookeries and hauling grounds

Sea otter concentration areas

Kelp beds

Foreign fishing areas

Archeological sites

The model accounts for seasonal vulnerability by associating with each target a vector specifying "home" 

or "away" for each month. When a simulated trajectory crosses a cell which the target matrix indicates 

may be occupied by a target, program SPILL checks 

to see if the target is home before registering a hit. 

Figure 2 shows a blue crab migration route in the Gulf 

of Mexico. A spill crossing this path might be assumed 

to not affect the crabs at times other than the migratory period. In assessing risk to migrating blue crabs 

from proposed offshore oil production in this area, hits 

on migrating crabs were recorded only when simulated 

spills contacted this path from September through 

February.

Modeling seasonal vulnerability inevitably requires 

some degree of professional judgment since assumptions must be made about the longevity of oilspill impacts. For example, an oilspill hitting a beach in May 

could still affect recreation in June.

LAND SEGMENTS

The model uses a special accounting system for simulated spills which hit land. The land areas near proposed oil production sites can be arbitrarily divided into two independent sets of land segments, with each 

set containing up to 99 segments. When a simulated 

oilspill hits a cell containing land, program SPILL 

checks to see which land segment contains this cell. 

The number of simulated spills hitting the shore (broken down into time-to-shore categories) are counted 

and stored by land segment.

TABLE 2. Targets for a risk analysis in the Eastern Gulf of Mexico 

(from Wyant and Slack, 1978)

Coral areas

Manatee concentrations

Brown pelican rookeries

Wading or pelagic bird rookeries

Dusky seaside sparrow habitat

Marine turtle nesting areas

American alligator habitat

Mangroves or tidal marsh

Estuarine nursery areas

West Florida adult female blue crab migration route

West Florida blue crab larval transport route

Tortugas pink shrimp nursery grounds

Calico scallops

Oysters and bay scallops

Seagrass beds

Spiny lobster

Sandy beaches

Florida Straits

High density use shoreline

National register sites

Designated wildlife, natural, and conservation areas

Designated national wildlife areas

National marine and estuarine sanctuaries

Florida aquatic preserves

Designated shoreline, national, and State parks

Ports

Foreign islands

Figure 3 shows a typical division of the shoreline of 

an analysis area into 52 land segments. The example 

comes from a risk analysis for a proposed Eastern Gulf 

of Mexico offshore oil production area (Wyant and 

Slack, 1978).

Compact storage of land segment numbers corresponding to each grid cell is achieved by breaking down 

IBM computer words in the 480X480 array. The wordbreakdown method for overall targets was described 

earlier; the method for land segments differs, but is 

similar in principle. The computer time required to access land segment information during a trajectory run 

is much less than that required for targets, as the land 

segment array need be consulted only when land is hit. 

Program SEGMATRX inserts the land segment information into the model in the appropriate format.

A few examples will clarify how an analyst might use 

the land segment feature of the model. If the estimated 

overall spatial distribution of spills hitting shore is desired, one set of land segments can simply divide the 

shore into equal-length units; counts of simulated 

spills hitting each equal-length segment provide the 

necessary information. If risk analyses are needed for 

each individual political jurisdiction in the overall 

analysis area, the second set of land segments could 

divide the shore into counties or other political units.

A further advantage of land segments is that they 

allow consideration of risks to targets which may not 

have been included in the model runs. For example,
REPRESENTATIONS OF PHYSICAL DATA

ALASKA

GULF OF ALASKA

i i i

NAUTICAL MILES

FIGURE 1. Map showing a sample target in the Western Gulf of Alaska. Hatched areas indicate foreign fishing areas.

Rectangles are proposed lease tracts (Slack, et al, 1977).

suppose that after the model has been run, a shoreline 

species is added to endangered species lists. Risk to 

the species can be estimated by examining the land 

segments in which the species resides.

Finally, the model is not applicable in many bays and 

estuaries. In a risk analysis for the Mid-Atlantic coast 

(Slack and Wyant, 1978), simulated spills were not permitted to enter the Chesapeake or Delaware estuaries 

where the trajectory assumptions of the mdoel are not 

applicable. To count simulated spills which would have 

entered the bays, the bay entrances were treated as 

parts of the shoreline, and a land segment was associated with each bay entrance. Counts of simulated spills

hitting these land segments allowed analysis of risk to 

the bays as a whole without addressing the further 

problems of spill movements within the bays.

CHECKING TARGETS AND LAND SEGMENT DATA

The model is designed to allow treatment of extensive and intricate spatial information. In addition to 

creating computer storage and run-time problems, the 

size and complexity of the model's basic data structure 

creates validation problems. Inattention to errors in 

data input can often lead to disastrously misleading 

output. Given the time and tedium required for data
OILSPILL RISK ANALYSIS MODEL OF THE U.S. GEOLOGICAL SURVEY

A TL ANTIC 

OCEAN

DO.

P D 

G ULF of MEXICO

0 50 I I I I I I

NAUTICAL MILES

FIGURE 2. Map showing a sample target in the Eastern Gulf of Mexico. Hatched area indicates blue crab migration route.

Rectangles are proposed lease tracts (Wyant and Slack, 1978).

checking and the greater intellectual satisfaction of 

tinkering with the analytical specifications of the model, it is always tempting to pay too little attention to 

this possibility. The computer programs have been designed to make data checking as complete and convenient as possible, and to prompt modelers to thoroughly 

carry out this phase of an analysis. OBJECTS and 

other spatial data entry programs routinely provide 

diagnostic information such as the number of points in 

the overall grid system used to represent each target. 

A coded version of the array used to store the target 

locations is routinely printed in each run of OBJECTS. 

The most important checking routines, however, are 

graphical.

Computer graphics provide a powerful tool for quickly and fully examining complex spatial data. This tool 

is exploited throughout the data entry phases of a 

model run. Program DIGIPLOT plots each target as it 

resides on computer tape immediately after entry from 

a digitizer. (The target's location at this stage is stored 

as a string of x-y coordinates representing locations

along the boundary of the target area on a map laid on 

the plane of the digitizer table. See the next subsection, "Insertion of Spatial Data into the Model," for 

more detail.) Timely examination of freshly entered 

spatial data using DIGIPLOT speeds the data entry 

phase of a model run and prevents costly cascading of 

errors through subsequent programs.

When program OBJECTS has inserted target locations into the final grid system, program OBJPLOT 

produces plots such as figures 1 and 2. These plots 

allow quick appraisal of how faithfully and completely 

target location in the final coordinate system agrees 

with the target location on the original map. These 

plots also provide an immediate check on the correctness of the various map scalings, rotations, and projections required to combine spatial information from different maps and different sessions on the digitizer.

In addition to providing the key to thorough and economical data checking, these computer-graphics programs are an invaluable tool for communicating the 

content and output of the Survey model. Model results
REPRESENTATIONS OF PHYSICAL DATA

47

A TLANTIC

OCEAN

G ULF of MEXICO

33 r

50

.51

52

NAUTICAL MILES

FIGURE 3. Map showing a typical division of the shoreline into land segments (Wyant and Slack, 1978).

must often be presented to users from a variety of technical backgrounds. Pictures such as figures 1 and 2 are 

easily understood.

INSERTION OF SPATIAL DATA INTO THE MODEL

This subsection describes the mechanical details of 

inserting spatial data into the model.

Target location is originally provided on a map of 

part of the overall analysis area. Each map must have 

a pair of reference points corresponding to a pair of reference points on the overall map of the area.

The map is laid on a digitizer table, and the outline of 

a target is traced with the digitizer's electrical crosshairs. This converts the image of the outline to a sequence of points expressed in digitizer table coordinates. The digitizer stores this sequence of coordinates 

on computer tape.

Program DIGIPRE screens the digitized locations 

of reference points, targets, and shorelines and stores 

them on a direct access disk pack in a form accessible

to program OBJECTS. Program DIGIPLOT creates 

diagnostic plots of target locations from these disk 

files to check against the original maps.

Several options are available for entering spatial 

data. Correct use of the options speeds the entry process and simplifies data organization and storage. Programs DIGIPLOT and OBJECTS automatically 

check for large gaps in the point sequences representing target outlines. Thus, the outline of a target with 

many discrete subareas, such as an island chain, can be 

traced on the digitizer table and the model will automatically recognize the individual islands. Targets 

representable as polygons can be entered simply by 

digitizing the polygon vertices; they need not be traced 

in their entirety. Some targets can also be entered as 

isolated points, but this presents some theoretical difficulties, since oilspills are also represented as points.

Land segments are entered much the same as polygonal targets. The order in which the polygon vertices 

are digitized is important a specific order is needed to
8 OILSPILL RISK ANALYSIS MODEL OF THE U.S. GEOLOGICAL SURVEY

let subsequent programs produce plots like figure 3. 

Land segments are stored somewhat differently from 

overall targets and are processed by programs 

DIGICOPY and SEGMATRX rather than OBJECTS 

and DIGIPRES.

OBJECTS performs several functions. The sequence 

of points representing a target outline on the digitizer 

table is scaled, rotated, and projected into the final 

coordinate system. The grid cells occupied by these 

points are noted, and the grid outline of the target is 

completely connected using subroutines GETLIM, 

NBR, and TRACK. The grid cells inside the outline are 

then found using subroutine FILL. Grid locations of 

the targets, or segments, are compactly stored in arrays using the compaction methods described previously. The arrays are then stored on a direct access 

disk in such a way that they are accessible to the trajectory program SPILL via subroutine NEWBLK's 

paging system. They are also accessed by program OBJPLOT, to produce drawings of the target locations, as 

in figures 1 and 2. SEGMATRX performs an identical 

function for land segments.

PAGING SYSTEM FOR LARGE ARRAYS

The 480 X 480 arrays identifying targets and land 

segments occupy almost 1.4 megabytes of storage. 

Since only a small portion of each array is needed at 

any time, a paging system has proven economical in reducing computer core requirements.

Each large array is dividied into 30X30 blocks, 

which are stored as direct-access records on a disk. The 

paging system will retain the most recently used 

blocks in core, and access the others as needed. A further refinement for the array of targets is to construct 

a 16 X16 array (one element for each block of the target array) that indicates presence or absence of target 

categories in each of the blocks. Thus, by checking the 

smaller array, the computer can determine whether or 

not it is necessary to read a block of the larger array.

WINDS

The subsections which follow explain how wind data 

is put into a form that can be used for oilspill simulation. Movement of oil under the influence of wind is 

covered in "Oilspill Trajectory Simulation."

STOCHASTIC MODEL OF WIND DATA

The variation in the wind is represented as a firstorder Markov process. That is, the wind in one timestep is a random function of the wind in the previous 

time-step. This reflects one's experience that if the

wind is presently out of the north at 5 knots, the wind 

3 hours or so from now will quite likely be the same, 

though there is a smaller chance of a large wind shift. 

A probability transition matrix, constructed from the 

historic wind record is used to model this Markov process. An example of a wind transition matrix is shown 

in figure 4, and provides for 41 wind velocity states (8 

directions time 5 speed classes, plus the calm state). 

The elements of this matrix are the probabilities that a 

particular wind velocity will be succeeded by another 

wind velocity in the next time step. For example, if the 

wind is now from the north at 10 knots, row 2 of the 

matrix shows there is a 22 percent chance that, 3 hours 

from now, the wind will still be from the north at 10 

knots, and that there is a 9 percent chance it will be 

from the northwest at 5 knots. If the present state of 

the wind is i, then the next wind state, j, can be randomly chosen by procedures described in the subsection, "Constructing Wind Transition Matrices: Programs RAWWIND, LISTWIND, and WINDTRAN."

Program WINDTRAN constructs the wind transition matrix from the historic record at a wind station. 

The resulting matrix is a description of the frequencies 

of wind velocity transitions that have occurred during 

the period of record. Probabilities of transitions not occurring in the record are assigned the value of zero.

There is an important difference between sampling 

winds from a Markov transition matrix constructed in 

this manner, and simply reading the historic wind record with randomly selected starting days. Although 

neither technique will model an individual transition 

which has not occurred in the past, the Markov process 

model can yield sequences of transitions which have 

not been observed in the historic record. Since a 30-day 

oilspill trajectory, with winds sampled every 3 hours, 

will involve a sequence of 240 wind transitions, a far 

greater number of sequences can be sampled from the 

transition matrix without repetition than is available 

from reading the historic record. In effect the difference is that reading solely from the historic record assumes that only wind transition sequences that have 

occurred in the past can happen in the future, whereas 

sampling from a transition matrix assumes that the sequence of wind transitions observed in the historic record is only a sampling of some underlying distribution. Considering that usually only 5 to 10 years of historic record is available, and that the oilspill simulation is to represent an exploration and production period of 20 to 25 years, this assumption seems appropriate.

The ideal wind data would be obtained from longterm weather stations located in the area of interest 

measuring wind velocity at the surface of the ocean. 

Unfortunately, there are few permanent stations at
REPRESENTATIONS OF PHYSICAL DATA

NEXT WIND IS OUT OF ,,, AT SPEED ... IN PERCENT.

LAST 

FROM

OIR

CALM

N

N

N

N

N

NE

NE

NE

NE

NE

E

E

E

E

E

SE

SE

SE

SE

SE

S

S

S

S

S

Sw

sw

SM

SW

SW

M

w

w

w

w

NW

NW

NW

NW

NW

CALM N 

510152025

SPO

0

5

10

15

20

25

5

10

15

20

25

5

10

15

20

25

5

10

15

20

25

5

10

15

20

25

5

10

15

20

25

5

10

15

20

25

5

10

15

20

25

27

28

10

6

50

0

26

22

0

0

0

11

7

0

0

0

13

2

0

0

0

26

9

0

0

0

27

20

0

0

0

33

15

0

7

0

27

7

2

0

0

2 1

1013

1522

Oil

050

0 0

6 1

022

o««

0 0

0 0

1 0

0 0

0 0

0 0

0 0

3 0

0 0

0 0

0 0

0 0

3 0

0 0

0 0

0 0

0 0

0 2

3 0

0 0

0 0

U 0

1 0

0 1

0 0

0 0

050

6 1

* 2

5 2

7 7

0 0

0 0

1 0

6 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

1 0

0 0

7 0

0 0

1 0

1 0

1 0

0 0

0 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

NE 

510152025

3 0

6 1

9 1

0 0

0 0

0 0

9 1

0 0

0 0

0 0

0 0

1 0

2 0

0 4

0 0

0 0

2 0

0 0

3 0

0 0

0 0

3 0

0 0

0 0

0 0

0 0

5 0

0 0

0 0

0 0

0 0

0 0

1 0

3 0

0 0

0 0

2 0

1 0

1 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

E 

510152025

34 3 0

11 0 0

600

000

000

000

16 4 0

Oil 0

000

000

000

5011 0

4027 4

42824

01325

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

013

0 0333333

46 6 1

972

10 3 8

008

000

24 0 0

300

600

000

000

lb 0 0

10 0 0

13 0 0

000

000

900

300

000

000

000

10 0 0

200

QUO

000

000

0 0

0 0

8 0

8 4

0 0

0 0

o u

o u

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

SE 

510152025

40000

20000

00000

00000

00000

00000

81000

Oil 0 0 0

00000

00000

00000

41000

52100

0 41612 4

0 0132513

00000

10 4 1 0 0

112018 7 2

0102618 3

0 81923 8

014292914

30300

93300

66000

014 029 0

00000

60000

00000

13 0 0 0 0

00000

00000

30000

10000

00000

00000

00000

10000

00000

00000

00000

00000

S 

510152025

1 0

0 0

0 0

0 0

0 0

0 0

0 0

Oil

0 0

0 0

0 0

0 1

1 0

0 0

0 0

0 0

1 2

4 2

0 3

0 0

014

15 6

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

4 0

ti 0

412

o o

0 3

121212 3

61919 0

0 0

0 0

2 5

7 7

13 0

033

0 0

1 0

1 0

0 0

0 0

0 0

0 0

0 0

0 U

0 0

0 0

029

0 0

3 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

SW 

510152025

1

1

1

6

0

0

1

0

0

0

0

1

0

0

0

0

1

2

0

0

0

9

9

0 0

0 0

1 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

2 0

3 0

0 0

0 0

6 0

9 3

01313

0

0

8

23

0 0

0 0

5 0

7 0

02513

0

0

2

3

0

0

0

1

2

0

0

0

033

0 0

1 0

1 1

3 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

2 0

0 0

4 0

0 0

0 0

0 0

6 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

w

510152025

6 4

0 2

0 0

0 0

0 0

0 0

8 4

11 0

0 0

0 0

0 0

8 5

4 2

0 0

0 0

0 0

4 1

2 0

0 0

0 0

0 0

0 0

6 9

0 0

1 0

0 0

3 0

6 0

0 0

0 0

1 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

2 0

0 0

0 4

0 0

0 0

0 0

6 0

01414 0

0 0

8 6

313

1313

033

0 0

23 7

2517

013

U 0

0 0

0 0

0 0

0 0

0 0

1 0

2 2

8 3

0 714 7

0 0

8 4

7 4

2 0

0 0

0 0

050

1 0

1 0

2 0

2 3

025

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

7

0

0

0

0

0

0

NW 

510152025

73100

1211 400

412 7 1 0

111144 6 0

00000

00000

35600

11 0 0 0 0

00000

00000

00000

22000

21000

40000

00000

00000

42100

00000

00000

ooooo

00000

ooooo

ooooo

ooooo

ooooo

ooooo

53000

30300

ooooo

ooooo

ooooo

10 7 2 1 0

1513 3 1 0

11292111 0

7 72114 0

OOOOO

1914 4 1 0

203015 2 0

11293110 0

2133626 3

0 0 02550

OVERALL 16 22000 20000 24 5100 31110 11000 11000 85100 88510

* Indicates greater than 99 percent probability

FIGURE 4. An example of a wind transition matrix. A 3-hour wind transition matrix for winter at Point Arguello, California (See also fig. 5).

sea a handful of lightships and "Texas towers." 

Thus, it is necessary to find other sources of data that 

can reasonably portray winds at sea.

Wind data recorded by ships have the advantage of 

originating from the area of interest and of (usually) 

being measured near the sea surface. For determining 

average wind conditions, these data are quite useful, 

although they may reflect the tendency of ships to

avoid adverse weather. However, because the data are 

not collected on a consistent, regular basis, they are 

not suitable for calculating wind transition probabilities.

Permanent weather stations provide long-term, continuous records of winds. It is not difficult to find stations that sample wind velocity every three hours and 

have been in operation for over five years. Such data
10 OILSPILL RISK ANALYSIS MODEL OF THE U.S. GEOLOGICAL SURVEY

are needed for constructing wind-transition matrices. 

However, permanent weather stations are usually located onshore (often at airports), away from the areas of 

oilspill interest, and may also be influenced by topographic effects, such as mountains.

The model combines the advantages of both types of 

data by comparing averaged ships' data, such as wind 

roses, for different parts of the study area, with the 

same data for permanent weather stations. In this 

manner, each part of the study area can be associated 

with the permanent weather station that most closely 

matches the ships' data in that region. Although this 

does not necessarily mean that the wind transition 

data are exactly the same, it appears to be the best 

that can be done with the available data.

CONSTRUCTING WIND TRANSITION MATRICES: 

PROGRAM RAWWIND, LISTWIND, AND WINDTRAN

Data collected by permanent weather stations are 

available on magnetic tapes in standardized formats. 

Program RAWWIND reads these tapes, excludes extraneous data, and stores the wind record for each day 

on a disk file in a compact, unformatted form. Once 

processed onto the disk file there is no further need for 

the rather cumbersome weather station tapes.

Wind sampling procedures may differ among weather stations. For example, some collect data at hourly 

intervals, others at six-hour intervals; some only collect data in the daytime. To ensure that the weather 

station record is suitable for sampling wind transitions, program LISTWIND provides a compact printout of the wind data. By examining this printout, the 

analyst can decide upon the appropriate course for further wind data analysis.

Once the wind data for a station are stored on a disk 

file in suitable form, a wind transition probability matrix is constructed for each season by program WINDTRAN. This program reads the wind record at a specified sampling interval, and classifies the wind into one 

of 41 wind velocity classes (eight directions times five 

speeds, plus the calm). It then looks ahead to the next 

sample to determine how the wind has changed. When 

data input is completed, WINDTRAN computes a 

wind transition probability matrix, with elements, aj / 

as follows:

n^-X 10.000 if 4i 7

41 k =

41

0, ifZ iH* = 0,

k = 1 

where

a; / = probability (times 10,000) that, if the wind is in

state i, the next sample will be in state 7,

HIJ = observed number of transitions observed from 

state i to state j.

Thus, if R is a random number between 1 and 10,000, 

and, the starting state is i, the next state, k, can be 

found by summing the elements of row i such that:

k + l

Z a; j>R> 7 = 1 l>3

k

Z a; ,- 

7 = 1 l'J

(2)

WINDTRAN must perform several other operations, 

in addition to constructing the transition matrices. 

First, it calculates an average speed and direction for 

all of the observations within each velocity class. This 

is done because selection of the classes is somewhat arbitrary, and given a finite number of samples, could introduce a bias in the simulated wind record. By using 

actual averages for each class (rather than only the nominal speed and direction), a simulated wind record 

should, in the long run, reproduce the averages of the 

observed winds. Thus the nominal designation of a 

class as "from the north at 5 knots" may actually 

mean "from the direction 2 degrees at 5.3 knots." Of 

course, as the number of observations increases, the 

two will become more and more alike.

The assumed wind drift angle (usually 20 degrees 

clockwise, in the northern hemisphere) is added directly to the average direction determined for each category. Then, the average wind vector is divided into x 

and y components of the coordinate system. Thus, the 

velocity class is found by a random sampling of the 

wind transition probability matrix, and the wind vectors for computing oilspill movement are found in a 

table for the appropriate class.

The final operation of \yiNDTRAN is performed to 

enhance the computational speed of later programs. 

Equation 2, which uses the ordinary transition matrix, 

is unnecessarily cumbersome for fast calculations: one 

must try an average of 20 values of k for each solution. 

Greater speed is attained by sorting and summing 

across the rows so that, in effect, the most likely transitions are sampled first. An additional matrix is 

needed as an index to the sorting, but since most wind 

transition probability matrices are strongly diagonal, 

the net result allows a much faster search.

The sorted wind transition probability matrices,, 

along with the corresponding indices, and the x and y 

oilspill movement vectors, are all stored on a disk file.

DEFINING WIND ZONES: PROGRAM WINDZONE

As explained earlier, winds in different parts of the 

study area may be simulated using the records of different permanent weather stations. Program WIND-
OILSPILL TRAJECTORY SIMULATION 11

ZONE assigns to each 10 X 10 block of grid cells a 

selected wind station number. By reading the wind station number, program SPILL can find the correct wind 

station to use for any location. Up to six sets of wind 

transition probability matrices, constructed from the 

records of six permanent weather stations, are permitted. Figure 5 shows an example of wind zones used 

for OCS Lease Sale 48; this particular analysis used 

four weather stations.

CURRENTS

Ocean currents are represented in the model as varying from month to month in a deterministic fashion. 

This is in contrast to the winds, which vary randomly 

over a relatively short time period. Spatial variation of 

currents is incorporated by dividing the study area into as many as 600 subareas, and assigning monthly 

current vectors to each of these subareas.

The model does not actually model ocean currents 

but utilizes a current field determined by other means. 

Input data for currents, whether derived from mathematical models or from direct measurements, must 

conform to the assumption (made in the preceding sections) that winds and currents are uncorrelated within 

a given month. Therefore, the current field used for the 

model is the baroclinic current, and all wind-induced 

currents must be represented with the wind data.

Tidal currents are also not included in the model. 

Generally, the waters in which tides are an important 

transport mechanism are not within the model's intended scope of analysis.

CURRENT DATA ENTRY AND STORAGE: 

PROGRAMS CURPOLY AND CURMATRX

Current data are made available to the model from 

maps of the study area showing the current field for 

each month. The study area is then divided into subareas or polygons, with 600 the maximum number. 

Each polygon is assigned a current vector for every 

month. The polygons are, therefore, a finite-element 

representation of the current field.

The polygon configuration must be able to adequately characterize the overall monthly current fields with 

the fewest possible polygons. At present, the judgments of the analysts and modelers are the sole determinants of the polygon configuration for each analysis. 

No mechanical polygon construction routine exists. 

Figure 6 shows a monthly current field of 518 current 

vectors used by the model in a run for a southern 

California risk analysis (see Slack, Wyant, and Lanfear, 1978); figure 7 shows the 518 polygons.

The vertices of the current polygons are first digitized in the same manner as land segment polygons 

(see "Land and Targets"). Program DIGICOPY transfers the digitized information into direct access disk 

datasets. Program CURPOLY combines the information in these datasets into a single current polygon file. 

Program CURMATRX then carries out the final processing steps to make current data accessible to program SPILL: (1) The polygon vertices are scaled, rotated, and projected from digitizer table coordinates to 

the final model coordinate system; (2) a 480X480 

2-byte integer array is filled with the current polygon 

identification numbers corresponding to each grid cell; 

(3) this array is stored on a disk file accessible to program SPILL's paging system; (4) the monthly current 

velocities associated with each polygon are read from 

cards and stored in another disk file. CURMATRX also generates diagnostic plots for data checking and a 

printout of the grid system showing the current polygon associated with each cell.

CURRENT DATA CHECKING

The special requirements of the model and the occasional need to obtain current data from several sources 

necessitate the translation of large amounts of current 

data to the appropriate format at the beginning of every analysis. Several programs of the model enable 

quick and thorough graphical checking of the final-format current data to detect translation errors. Graphics 

are especially effective in detecting major errors. Programs CURPLOT and CURMATRX provide plots of 

polygon locations, plots of spatial current fields, by 

month, and diagnostic printouts of the final 480 X 480 

current grid.

OILSPILL TRAJECTORY SIMULATION

MONTE CARLO SIMULATIONS

OF OILSPILL TRAJECTORIES

For each selected launch point, a large number of hypothetical oilspills are released at randomly chosen 

days within the year and are moved about by randomly 

sampled winds and currents. With sufficient trials, the 

statistical behavior of the trial spills will approximate 

the statistical behavior of spills integrated over all possible combinations of release times, winds, and currents.

The model analyzes oilspill trajectories from a set of 

potential launch points which are chosen to represent 

different proposed oil production sites in the OCS lease 

area and proposed transportation routes. A total of 

100 launch points may be selected. From each launch 

point, 500 hypothetical oilspills are simulated for each 

season of the year, resulting in as many as 200,000 

simulated oilspills for a model run. The next section 

shows how these simulation results are further analyzed to determine risks from various parts of the lease 

area.
12 OILSPILL RISK ANALYSIS MODEL OF THE U.S. G'EOLOGICAL SURVEY

FRANCISCO

CALIFORNIA

PT ARGUELLO

terv^ ^hk^^r 'O

§

SAN NICHOLAS

PA CIFIC O CEAN MEXICO

0 50 i i i i i i

NAUTICAL MILES

FIGURE 5. Wind zones used for OCS Lease Sale 48 southern California. Four weather stations were used instead of six. 

Rectangles are proposed lease tracts (Slack, Wyant, and Lanfear, 1978).

A single launch point may adequately portray a 

group of proposed lease tracts, but additional points 

are often needed to represent pipelines and tanker 

routes. An option of the model allows a launch point to

be specified as a straight line, rather than a single 

point; the 500 spills per season are started from 100 

uniformly spaced locations along the line, 5 spills at 

each location.
OILSPILL TRAJECTORY SIMULATION 13

)SAN FRANCISCO

CALIFORNIA

LOS ANGELES

SAN DIEGO

PACIFIC OCEAN MEXICO

0 50 i i i i i T

NAUTICAL MILES

1 KNOT IS

FIGURE 6. Monthly current field for southern California (March). Rectangles are proposed lease tracts. Lines represent

current vectors (Slack, Wyant, and Lanfear, 1978).

EFFECTS OF WIND AND CURRENT 

ON OILSPILL MOVEMENT

The effects of wind on a parcel of oil flowing on the 

sea surface have been studied by a number of investigators (Murray and others, 1970; Murty and 

Khandekar, 1973; Alien and Thanarajah, 1977; Phillips

and Groseva, 1977; Stolzenbach and others, 1977; 

Zilitinkevich, 1978). There appears to be only partial 

agreement on the general theory of wind-induced oilspill movement, probably because of the complexity of 

the subject. Winds may transport oilspills through 

wind-generated currents, wind-induced waves, and by
14 OILSPILL RISK ANALYSIS MODEL OF THE U.S. GEOLOGICAL SURVEY

CALIFORNIA

FIGURE 7. Current polygons for southern California developed from the monthly current field. Rectangles not 

containing a vector are proposed lease tracts (Slack, Wyant, and Lanfear, 1978).

direct wind shear; these effects can combine in different ways, depending on characteristics of the oil, sea 

conditions, and ocean-bottom topography.

Despite the theoretical difficulties, an empirical approach to predicting oilspill movement has proved 

quite successful in practical trajectory modeling. First 

described by Smith (1968) in a study of the Torrey Canyon spill off the coast of Great Britain, the method requires the following simple assumptions:

 The effects of wind and current on the oilspill act 

independently, and can thus be described as a 

simple vector sum of velocities.

 The wind vector is a constant small fraction of 

the wind speed, but the direction of oilspill mo-
OILSPILL TRAJECTORY SIMULATION 15

tion induced by the wind is at a nonzero angle

to the direction of the wind due to Coriolis

forces. 

 The current vector is equal to the current velocity.

Regarding the second assumption, the wind vector 

has been estimated empirically to equal 3.5 percent of 

the wind speed with a drift angle of 20 degrees to the 

right (clockwise) of the wind direction for the northern 

hemisphere (Smith, 1968; Stolzenbach and others, 

1977).

The independence of the effects of wind and current 

allows the forces to be calculated separately and the resultant motion of the oil to be taken as the vector sum. 

This requires, of course, that the current field be free of 

wind effects. Data on currents for past trajectory analyses using the model have come from many sources. 

Results of drift studies and the output of computer 

models have been used. Precise assessments of the 

validity of either drift study results or mathematical 

model outputs are hard to come by, but something can 

usually be said about the sensitivity of a set of oilspill 

risk analysis model results to assumptions regarding 

currents. More exactly, it is important throughout an 

analysis to remain aware of whether oilspill movement 

would be current-dominated or wind-dominated. Often, 

dominance differs both seasonally and spatially.

Figure 8 contrasts spill movements dominated by 

each mechanism. The figure shows 10 simulated trajectories launched from a point in a proposed oil production area off the Mid-Atlantic coast (see Slack and 

Wyant, 1978). For this area as a whole, average wind 

speed is 12.3 knots (based on lightship data). Assuming winds move oilspills at about 3.5 percent of the 

winds' own velocities, the winds in this area would, on 

the average, induce a 0.43 knot speed in spill movements. The currents in the immediate vicinity of the 

proposed production area are weak 0.1 to 0.3 

knots and the meanderings in simulated trajectories 

induced by shifts in the winds can be readily seen in 

the figure. The Gulf Stream runs to the east of the proposed production area, with currents at 1.0-2.0 knots. 

As simulated spills leave the lease area to the east, currents dominate over winds in influencing movement. 

Thus, the simulated spills in the eastern portion of the 

area move rapidly eastward into the Atlantic Ocean, 

with little meandering.

OILSPILL MOVEMENT

Program SPILL simulates oilspill movement as a 

series of displacements over finite time-intervals. For 

each time step in the duration of a hypothetical spill, 

two vectors one representing the effect of the wind

and the other that of the current are summed to obtain the displacement of the spill's center of mass. The 

spill is then moved in a straight line between its old 

and new grid coordinates as illustrated in figure 9, and 

any cells through which the spill passes are checked for 

the presence of targets. The tracking of a hypothetical 

spill continues in this manner until a time limit (usually 30 or 60 days) is exceeded, or until the spill contacts 

land or leaves the area being modeled.

The choice of the time step is based on the sizes of 

the current polygons, the persistence of the wind data, 

and practical limits for computer run time. Since a current vector is selected only at the beginning of a time 

step, a time step short enough to consider the smaller 

current polygons must be chosen, or they will be 

skipped over and ignored. Assuming a spill movement 

speed of 0.5 to 1.0 m/s, a 3-hour time step usually fulfills this condition; where current polygons are larger, 

a 6-hour time step may be satisfactory. A 3- to 6-hour 

time step also appears to adequately characterize the 

wind data in that it makes the model sensitive to the 

variability in synoptic weather patterns. Finally, a 

3-hour time step is a realistic limitation considering 

the computational speed of program SPILL using existing Geological Survey computer facilities.

Although program SPILL's function moving a 

simulated oilspill through cells, checking each cell for 

targets, and counting hits on the targets is simple in concept, it is a tedious and time-consuming task. A detailed explanation of how program SPILL accomplishes this, using a variety of programming techniques to increase its running speed, is not included in 

this paper. To understand the probability calculations, 

however, it is important to know the rules used for recording contacts (hits) of simulated oilspills with the 

targets. These rules apply to each simulated oil spill:

 The spill may only be designated as hitting or 

missing each of the 31 target classes; multiple 

hits on the same target class count as only one 

hit.

 SPILL automatically determines which months 

a target is vulnerable to oilspill damage, and 

counts hits only during these months.

 Upon first contact of the spill with each target 

class, the simulated age of the spill is recorded.

 If a spill contacts a cell containing land, its simulation is terminated, and the land segment 

code of that cell is noted; thus the spill may hit 

no more than one land segment in each set of 

land segments.

 If the spill moves off of the grid, its simulation is 

terminated and the direction in which it left 

the grid is recorded.
16 OILSPILL RISK ANALYSIS MODEL OF THE U.S. GEOLOGICAL SURVEY

ATLANTIC OCEAN

FIGURE 8. Example oilspill trajectories for a spill site (P4) near the center of the proposed Mid-Atlantic (OCS Lease Sale 49) lease 

area: summer conditions. Number on trajectory is the time to the end point in days. (Slack and Wyant, 1978.)
RISK CALCULATION 17

 If a spill continues beyond a fixed time limit 

(usually 30 or 60 days), it is assumed to have 

decayed, and its simulation is terminated.

 The final grid location of the spill is recorded.

Program SPILL produces a record, on a disk file, of 

the behavior of each hypothetical spill. A summary of 

SPILL's output is created by program SUMMARY, 

which shows the results in groups of 100 spills, so that 

the variability of the Monte Carlo simulation can be 

checked.

A variation of SPILL (identical to the main program, 

but containing plotting subroutines) is used to produce 

graphical displays of trajectory runs. Graphical displays help the analyst ensure that simulated spills behave in a logical manner, and effectively detect errors 

such as improper scaling factors and reversed wind or 

current fields. These displays, such as those shown in 

figures 8 and 10 are also useful as examples of the performance of the model. Conclusions about oilspill behavior from such displays should be cautioned against, 

since a figure showing 10 spills represents only 0.5 percent of the 2000 spills launched. While average probabilities of hits by oilspills is a meaningful concept, 

there is no such thing as an "average" trajectory.

A paging system for storing and retrieving the large 

matrices containing current and land segment data 

holds down the size of SPILL. Even so, its 500-kilobyte size makes it the second largest of the model's 

programs. For a large OCS lease sale analysis, SPILL 

may require more computer operating time than all of 

the model's other programs combined. Because of its 

long running time, SPILL is usually run in 5 to 20 independent jobs, so that no one job uses more than onehalf hour of central processing unit (CPU) time. The 

output files of all the jobs are concatenated to form the 

complete output file. On an IBM 370/155 computer, 

SPILL can operate at a speed of 1 millisecond per time 

step, or about 1A second for a single 30-day oilspill 

simulation.

RISK CALCULATION 

CONDITIONAL PROBABILITIES

Program SPILL records, on a disk file, data about 

the trajectories of 2,000 hypothetical oilspills from 

each launch point and the contacts made by these trajectories on targets and land segments. SPILL does 

not perform any analysis or interpretation of this data; 

summations and statistical analyses are performed by 

a subsequent set of programs. These programs determine the likelihood, or conditional probability, that

certain events, such as contact with targets or land 

segments, will occur if an oilspill occurs at a given 

launch point.

Separating the conditional probability analysis from 

the Monte Carlo simulation permits the large and timeconsuming program SPILL to remain relatively 

straightforward, while its output can be tailored to 

user requirements with small, easily modified programs. Two programs, HITPROB and LANDSEG, 

are used to calculate the conditional probabilities of 

spills contacting targets and land segments, respectively. A third program, FIRSTPAS, analyzes the 

travel times oilspills need to reach targets. All three 

programs operate in a similar manner, scanning the 

disk output of SPILL to review the results of each trajectory run, and selecting and tabulating the necessary 

information.

CONDITIONAL PROBABILITIES OF CONTACTING TARGETS: 

PROGRAM HITPROB

Program HITPROB calculates the probability that, 

if an oilspill occurs at a given launch point, it will, within a specified period of time, contact a target. Conditional probabilities are calculated for each launch point 

for oilspills with maximum ages of 3, 10, 30, and 60 

days. Typical output from HITPROB is shown in table 

3; this same information is stored on a disk for use by 

program NU, which calculates overall risks. (See section on "Probability that an Oilspill will Occur and 

Contact a Target," for elaboration of program NU). 

Since SPILL records a target as "hit" only if contact 

occurs during a month in which it is vulnerable to oilspill damage, the condition probabilities automatically 

reflect any seasonal vulnerability.

It is important to recognize that the conditional probabilities calculated by program HITPROB refer to 

each target as a whole and imply nothing about the distribution of risk among any subdivision of that target. 

For example, the target "sandy beaches" may extend 

for several hundred miles of coastline, and risks to particular beaches may differ. Program HITPROB would 

only calculate the conditional probability that an oilspill originating at a given point would land on a sandy 

beach somewhere in the study area; it would tell 

nothing about the likelihood of contacting a specific 

beach (except that it is less than or equal to that of contacting "sandy beaches"). If such ditferentation is desired, then each item should be defined as a single target or the land segment feature of the model should be 

used.
18 OILSPILL RISK ANALYSIS MODEL OF THE U.S. GEOLOGICAL SURVEY

MOVEMENT OF OILSPILL DURING TIME 

STEP, FROM (Xt.Yt) TO (Xt+At, Yt+At)

_ T " PATH FOR CHECKING CELLS

SHADED CELLS ARE CHECKED DURING 

TIME STEP

FIGURE 9. Movement of a hypothetical oilspill through the model's 

grid system during one time step.

CONDITIONAL PROBABILITIES OF CONTACTING 

LAND SEGMENTS: PROGRAM LANDSEG

Program LANDSEG calculates the probability that, 

if an oilspill occurs at a given launch point, it will, within a specified period of time, contact a particular segment of coastline. As in HITPROB, conditional probabilities are calculated for each launch point for oilspills 

with maximum ages of 3, 10, 30, and 60 days. Each of 

the two sets of land segments is processed independently, using two slightly different versions of 

LANDSEG, called LANDSEG 1 and LANDSEG 2. 

Typical output from LANDSEG is shown in table 4; 

the same type of information is stored on a disk for use 

by program NU, which calculates overall risks.

Identification of land segments does not explicity account for spreading of the oil. Although a large oilspill, 

in reality, could affect several land segments, a "hit" is 

scored on only one; the user must examine neighboring 

segments, as well as oilspill travel times, and separately calculate the possible extent of spreading.

TRAVEL TIMES FOR OILSPILLS CONTACTING TARGETS: 

PROGRAM FIRSTPAS

Program FIRSTPAS calculates the average, minimum and maximum times-of-travel for oilspills occurring at a given launch point to make first contact with 

a target. This tabulation, an example of which is shown 

in table 5, is presented by season as well as for the entire year. Spills which do not contact a target are not 

included in the statistics for that target.

Program FIRSTPAS was, in earlier versions of the 

model, the only means of accounting for oilspill age. 

When HITPROB was revised to present its results for 

spills of different travel times, FIRSTPAS became 

partially obsolete. However, it has still proven to be a 

useful program for checking the behavior of the model 

and for helping to understand oilspill transport.

SPILL OCCURRENCE

This section describes how spill occurrence probabilities are estimated.

To construct the estimated probability distribution 

on spill occurrence for a fixed class of spills, certain 

simplifying assumptions must be used which may be 

unsatisfactory in particular instances. The forecasting 

method used in the model is sufficiently flexible for incorporation of new and specific assumptions, however.

The following were considered some desirable features of a spill occurrence forecasting method:

1. The method should include an estimate of the 

uncertainty in the forecast by providing a 

probability distribution rather than a predicted number of spills.

2. The method should be consistent with past observations and intuitively reasonable.

3. The dependence on past occurrence rates 

should be clear and explicit.

4. The method should be flexible; that is, changes 

in the assumptions concerning use of past occurrence rates should be easily accommodated, and the method should be easy to update as new data are accumulated.

SOME BASIC FEATURES OF SPILL 

OCCURRENCE FORECASTING

Forecasts of oilspill occurrence are made via a predicted probability distribution on the number of spills 

which might occur during the production life of a lease 

area. The predicted distributions are constructed using 

Bayesian methods to incorporate the uncertainty due 

to limited historic spill-incidence data. The appendix 

describes this method in detail.

Simple summary statistics to describe the frequency 

of spills expected to occur during the production life of 

a lease area must be chosen to reflect, as best as possible, the shape of the probability distribution. Considerable uncertainty in forecasting for a new offshore 

lease area is reflected in a predicted probability distri-
RISK CALCULATION 19

SAN FRANCISCO

CALIFORNIA

LOS ANGELES

NAUTICAL MILES

DIEGO

PACIFIC OCEAN MEXICO

so

FIGURE 10. Example oilspill trajectories for a spill site near southern California (OCS Lease Sale 48). Rectangles 

are proposed'lease tracts. Number on trajectory is the time to the end point in days (Slack, Wyant, and 

Lanfear, 1978).

bution with high variance, implying that one cannot 

forecast a single number of future spills with much 

confidence. Presenting the "expected number" of spill 

can be misleading, as a wide range of possible spill 

totals may be as likely to occur over the life of the lease 

area as the "expected number," which is the 

hypothetical average over many lease area lifetimes. 

Thus, model forecasts are presented in terms of the 

most likely number of spills based on the predicted 

probability distribution (in statistical terms, the mode

rather than the mean) as well as the predicted probability that one or more spills of a given size will occur in 

the lifetime of a lease area.

Spill occurrence forecasts are made separately for 

different spill-size categories. Oilspills of different 

magnitudes have different damage potentials, and 

may be expected to exhibit different statistical properties in their occurrence. The largest spills occur relatively rarely, but account for a large proportion of the 

total volume spilled. For example, the Argo Merchant
20 OILSPILL RISK ANALYSIS MODEL OF THE U.S. GEOLOGICAL SURVEY

TABLE 3. Example of typical output from program HITPROB, showing probabilities (expressed 

in percent chance) that an oilspill starting at a particular location will reach a certain target in 

30 days

[n, less than 0.5 percent chance; *, greater than 99.5 percent chance]

Hypothetical spill location

Target

Land

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

PI

3

1

n

2

n

n

n

n

n

1

n

n

n

n

n

3

n

n

n

1

n

3

n

n

n

n

n

n

n

P2

-E2

2

1

1

1

n

1

1

n

n

1

n

n

n

n

n

2

1

1

n

1

n

n

n

n

n

n

n

n

n

P3

-E3

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

P4

-E4

1

n

1

1

n

n

1

n

n

n

n

n

n

n

n

1

n

n

n

n

n

n

n

n

n

n

n

n

n

P5

1

n

1

1

n

n

n

n

n

n

n

n

n

n

n

1

n

n

n

n

n

n

n

n

n

n

n

n

n

P6

1

n

n

n

n

n

n

n

n

n

n

n

n

n

n

1

n

n

n

n

n

n

n

n

n

n

n

n

n

(launch point)

Tl

4

2

2

1

n

2

1

n

n

1

n

1

n

n

n

4

1

n

1

2

n

1

n

n

n

n

n

n

n

T2

11

3

5

4

n

4

2

n

n

1

1

3

n

n

n

12

3

1

3

4

n

n

n

n

n

n

n

n

3

T3

39

6

15

11

n

4

2

n

n

5

1

4

n

n

n

44

9

5

11

4

n

n

n

n

n

n

n

1

*

T4

5

3

5

4

2

n

2

2

n

2

n

n

1

n

n

6

1

2

1

2

n

n

1

n

n

n

n

n

n

T5

17

6

19

12

3

n

6

4

n

5

n

n

4

n

n

19

3

5

5

4

n

n

3

n

n

1

n

n

n

T6

2

1

2

1

n

n

1

n

n

1

n

n

n

n

n

3

1

1

n

1

n

n

n

n

n

n

n

n

n

T7

11

8

6

5

1

2

3

1

n

5

n

n

1

n

n

11

4

1

1

3

2

n

1

n

n

n

n

n

n

n - less than 0.5 percent chance

* - greater than 99.5 percent cbance

spilled 7.7 million gallons when it broke up off Nantucket in December 1976 (Grose and Mattson, eds., 

1977, p. 1); by comparison, the total volume spilled in 

1975 by U.S. tankers was 7.8 million gallons in 587 incidents (Stewart, 1976, p. 60). The largest spills have 

the most damage potential, and generally occur under 

different circumstances from smaller spills. Major 

blowouts of wells or complete ship breakups, for instance, are somewhat distinct from minor collisions or 

equipment malfunctions.

Spill occurrence forecasts are made separately for 

different types of spill sources tankers, pipelines, and 

platforms. It is reasonable to expect that spill occurrence rates will differ for the various modes of production and transport, and past data support this contention (see table 6). A principal use of the risk analysis 

model has been to help compare transportation mechanisms for given lease areas.

Continued accumulation of data may enable greater 

refinement of spill-source categorization in the future.

For example, tankers might be considered separately 

by age class (Stewart and Kennedy, 1978, p. 25), or 

deep-water production rigs with single-buoy moorings 

might be considered separately from rigid, near-shore 

structures. The exact approach taken for a given risk 

analysis should depend on available data, the precise 

concerns of the analysis, and how the model results are 

to be interpreted; the model can be straightforwardly 

applied, using the same methodology, programs, and 

reporting structures as for the present pipeline-tanker 

breakdown.

Spill occurrence forecasts, as well as any assessment 

of risk from a given development program, depend fundamentally on the estimated amount of oil to be produced in a lease area. First described by Devanney and 

Stewart (1974), the Bayesian methodology used to construct the probability distributions on spill occurrence 

utilizes past production and oilspill occurrence data 

and future production estimates in a straight-forward 

way. The following sections provide further details.
RISK CALCULATION 21

TABLE 4. Example of typical output from program LANDSEG, showing probabilities 

(expressed in percent chance) that an oilspill starting at a particular location will reach a 

certain land segment in 30 days

[n, less than 0.5 percent chance]

Hypothetical spill location

Land

Segment

Al

A2

A3

AA

A5

A6

A7

A8

A9

AID

All

A12

A13

A1A

A15

A16

A17

A18

A19

A20

A21

A22

A23

A2A

A25

A26

A27

A28

A29

A30

PI

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

1

1

n

P2

-E2

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

P3

-E3

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

PA

-EA

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

P5

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

P6

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

Tl

n

n

n

n

n

n

n

n

n

n

n

n

1

2

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

T2

n

n

n

n

n

n

n

n

n

n

n

n

1

6

2

n

n

2

n

n

n

n

n

n

n

n

n

n

n

n

T3

n

n

n

n

n

n

n

n

n

n

n

n

1

8

18

1

3

8

n

n

n

n

n

n

n

n

n

n

n

n

TA

n

n

1

n

n

n

1

1

1

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

T5

n

n

2

1

n

n

1

2

5

5

1

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

T6

n

n

n

n

n

n

n

n

n

n

n

n

1

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

T7

n

n

1

n

n

n

n

1

1

1

n

3

3

1

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n - less than 0.5 percent chance

PREDICTED PROBABILITY DISTRIBUTION FOR 

A FIXED CLASS OF SPILLS

This subsection describes the predicted probability 

distribution for spill occurrence within a fixed class of 

spills. A fixed class of spills consists of spills in a single 

size range (say, spills larger than a thousand barrels) 

originating from a single spill source (say, tankers).

A basic assumption of the method is that spills occur 

as a Poisson process, with volume of oil produced or 

handled as the exposure variable. (Other exposure variables can also be considered, as discussed in the next 

subsection.) That is, the probability P, of observing n 

spills in the course of handling t barrels of oil is

i* (3)

where A is the spill occurrence rate per unit exposure, 

(spills per million barrels of oil).

The Poisson assumption requires that spills occur independently of each other. One could clearly question 

this assumption if, for example, safety and inspection standards were improved as a result of a particular spill, several potentially subsequent spills might be 

averted. Nonetheless, there is evidence that a Poisson 

model for spill occurrence provides a reasonable approximation (see Stewart and Kennedy, 1978, p. 36).

The spill occurrence rate, A, is unknown. A Bayesian 

methodology, described in detail in appendix A, provides one way to weight the different possible values of 

A, given the past frequency of spill occurrence for a 

fixed class of spills by taking a weighted average of the 

distributions (equation 3) over different values ofA. If v 

is the number of past spills in the fixed spill class in the
22 OILSPILL RISK ANALYSIS MODEL OF THE U.S. GEOLOGICAL SURVEY

TABLE 5. Example of output from program FIRSTPAS. Average minimum, maximum, and mean timeof-travel for oilspills occurring at site PI to contact targets. Winter season and all seasons.

LAUNCH POINT: PI 

500 SPILLS PER SEASON

TARGET

1

2

3

4

5

6

7

8

9

10

11

12

13

14

TANNER AND CORTEZ BANKS

RANGER BANK

MAJOR MARKETFISH

MAJOR COM PELAGIC FISH

SALMON

ALRACORE

BO N I TO

TUUA

SUOKDFISH

COMMERCIAL SHELLFISH

SKABIRDS (APRIL-JUNE)

SEABIRDS (JULY-SEP)

SKABIRDS (OCT-DEC)

SEABIRDS (JAN-MAR)

NO.

64

3

296

97

0

0

0

0

0

192

0

0

0

129

WINTER

MIN

11

53

0

10

0

0

0

0

0

1

0

0

0

5

MAX

59

56

60

60

0

0

0

0

0

60

0

0

0

60

MEAN .

33.3

54.6

14.4

32.9

0.0

0.0

0.0 .

0.0

0.0

21.8

0.0

0.0

0.0

27.1

. . NO.

350

13

1626

441

11

372

. . 11

78

18

1189

400

42

1

141

ALL

MIN

6

35

0

5

7

3

5

15

34

1

1

0

19

4

MAX

59

56

60

60

20

60

56

60

58

60

19

58

19

60

MEAN

25.5

46.9

7.0

26.5

12.7

27.5

35.2

41.3

45.2

11.2

2.6

16.2

18.6

26.0

course of handling T barrels of oil, the estimated predicted probability that there will be n spills in the next t 

barrels handled is

(4)

nl(v-l)l(t+T)n +v

This is the negative binomial distribution with expectation

E=vtlr (5) 

and variance

V = J!(1+_L) (6)

T T

The probability of one or more spills is

Thus, the predicted probability distribution equation 

on spill occurrence for a fixed class of spills (a single 

spill-size range, a single spill-source category) incorporates the predicted volume to be handled, t, the past occurrence rate, (V/T), and the uncertainty which stems 

from the fact that (vt/r) is not likely to equal the true 

occurrence rate, A, exactly.

CHOOSING AN EXPOSURE VARIABLE

Fundamental to the spill occurrence forecasting 

method is the notion of an exposure variable. An exposure variable is some quantity related to oil production 

or transportation which has a precise statistical relationship to spill occurrence. In the past, the exposure

variable used in the model has been volume of oil handled. Predicted probability distributions have been 

constructed by utilizing past rates of spills per volume 

of oil handled and the projected volume of oil to be handled.

Other exposure variables could be used. In the case 

of tankers, for example, number of port calls and numbers of tanker years have been contemplated (Stewart, 

1976, p. 53, and Stewart and Kennedy, 1978, p. 23). 

The model described here permits the use of any exposure variable without major alteration of programs or 

other parts of the analysis.

An exposure variable should measure some aspect of 

oil production or oil transport such that for an amount 

of exposure t the probability of n spills occurring is 

given by the Poisson distribution:

(3)

re!

where A is the average rate of spill occurrence per unit 

exposure and t is the exposure. This implies the following technical assumptions:

 The mean and variance of spills for a given 

amount of exposure should be At.

 Spills must occur independently.

In practice, this relationship holds only approximately for any specific exposure variable, and it may 

be impossible to reject any of several alternative exposure variables simply on the basis of analysis of past 

data. Further criteria for choosing exposure variables 

are:
RISK CALCULATION 23

TABLE 6. Historic spill occurrence rates used in the Oilspill Risk Analysis Model

Spill 

source

Number of 

spills

Volume handled 

(millions of bbls)

Data 

source

Area 

covered

Time period 

covered

Model runs 

used in

Spills over 1,000 barrels

Platforms .......

Do ...........

Pipelines ........

Do ...........

Tankers .........

Do ...........

9

10

8

11

99

178

3,927

5,338

3,169

4,780

29,326

45,941

Devanney and Stewart

(1974, p. 89).

Stewart (1975, p. 32)

and personal commun.

May 1977.

Devanney and Stewart

(1974, p. 94).

Stewart (H975, p. 33)

and personal commun.

May 1977.

Devanney and Stewart

(1974, p. 49).

Stewart (1976, p. 66)

U.S.

U.S.

U.S.

U.S.

World

World

1964-72

1964-75

1964-72

1964-75

1969-72

1969-73

Before

May 1977.

After May

1977.

Before

May 1977.

After May

1977.

Before

May 1977.

After May

1977.

Spills under 1,000 barrels

Platforms and 

pipelines. 

Tankers......

1,230 4,396

624 1,412

Devanney and Stewart

(1974, p. 102). 

Devanney and Stewart

U.S. 

U.S.

1971-72 

1971-72

All. 

All.

 The exposure should be simple.

 It should not intuitively violate the preceding 

technical assumptions to any significant extent.

 It should be a quantity which is predictable in

the future.

The last criterion is particularly important in forecasting applications. If the analyst has an estimate of future production from a large area, but no specific information on how the area is to be developed, in terms of 

number of platforms, etc., then volume produced 

might be preferable as an exposure variable over platform-years, even if platform-years appear to be a better 

exposure variable based on past data.

How can a contemplated exposure variable be checked using past data? One-way is by testing the assumption that the mean and the variance are both equal to 

Xt. The linear relationship between the expected number of spills and the exposure variable suggests the use 

of least-squares regression techniques; weighted least 

squares should be used because the variance of the 

number of spills is not constant, and is also linearly related to exposure (see Draper and Smith, 1966, p. 77). 

Thus, if (TL T2 ,..... T^) are the exposures in regions (rlt 

r2 ,..... rn) during some year, and (vlf v2 ,. . . ., vfj are 

the respective numbers of spills observed, then a regression of VjV-Tj-vs. \/~TJ checks the first technical assumption. This gives SVJ/STJ as the true rate of spill occurrence per unit of exposure. The usual tests of a

regression fit can be used to evaluate the appropriateness of this assumption. Mean and variance equal to At 

is a necessary but not sufficient condition for the 

Poisson model. (Stewart and Kennedy, 1978, p. 60, present this point quite forcefully). Devanney and Stewart 

(1974, p. 45) give some examples of regression investigations where volume of oil handled is used as an 

exposure variable.

Occasionally it will be possible to test directly the 

Poisson assumption in its entirety. If there are numerous observations, each with the same exposure, then 

the associated numbers of spills represent independent 

observations from a single Poisson distribution, and 

the standard statistical tests for goodness-of-fit can be 

employed. A possible base is tanker spills, where a contemplated exposure variable is tanker-years. Every 

tanker which has been in service for the same time 

period will have the same exposure. Stewart and Kennedy (1978, p. 24) performed goodness-of-fit tests in 

this situation and felt the Poisson model to be acceptable.

In practice, however, these statistical testing procedures rarely demonstrate unequivocally that a given 

exposure variable is "correct." They provide one way 

to rank contemplated exposure variables based on past 

experience. The ultimate choice of an exposure variable 

will rest largely on the judgment of the analyst.

The regression work of Devanney and Stewart (1974, 

p. 26) indicated that volume of oil handled is at least a
24 OILSPILL RISK ANALYSIS MODEL OF THE U.S. GEOLOGICAL SURVEY

reasonable exposure variable. The variable is simple, 

bears a good intuitive connection to the number of 

spills, and is relatively easy to predict in advance within known error limits. Recently, though, Stewart and 

Kennedy (1978) have investigated the use of other exposure variables.

SPILL OCCURRENCE RATES AND EXPOSURE VARIABLES

The sources for the spill occurrence rates used in the 

model are Devanney and Stewart (1974), and Stewart 

(1975 and 1976). Those authors obtained data primarily from three sources: the Conservation Division of the 

U.S. Geological Survey, the U.S. Coast Guard, and a 

survey of world-wide major tanker spills in 1969-1972 

(Devanney and Stewart, 1974, p. 1). In the past, there 

have been many problems in screening and reconciling 

the information in these data sources; Stewart and 

Devanney have done much in this area and describe it 

in the above-cited reports. Table 6 gives the spill occurrence information used to date in runs of the model. 

The occurrence rates were used to construct predicted 

probability distributions on spill occurrence as described in the earlier subsection, "Predicted Probability 

Distribution for a Fixed Class of Spills." For small 

spills, pipelines and platform occurrences are lumped 

together due to the data base ambiguity concerning 

the precise division point between a platform and a 

pipeline spill.

TRANSPORTATION SCENARIOS

The previous section presented a method for constructing a probability distribution on spill occurrence. 

The next logical step is to show how site-specific details are applied to calculations of spill occurrence.

CONSTRUCTION OF TRANSPORTATION SCENARIOS: 

PROGRAM SCENARIO

The risks of oilspills resulting from OCS development do not arise solely from platform operations. 

Transporting oil to shore entails additional risks which 

can exceed the risks of extracting the oil. Therefore, 

each group of leasing tracts must be considered as part 

of an integrated production and transportation system; program SCENARIO provides a means of describing this system so that spill occurrence probability 

distributions can be calculated.

For each production site, a transportation route 

must be defined by linking together any of the launch 

points analyzed by program SPILL with destination 

points (see figure 11). The method of transport (that is, 

pipeline or tanker) must be specified for each transportation route segment. It is not necessary for the route 

to be strictly continuous, since this description is only

an approximation of an actual route. The modeler must 

use judgment in striking a balance between a precise 

route description and a reasonable computational effort, and should be watchful against specifying a transportation route more detailed than justified by the 

resolution of the model. Figure 11 shows how oil produced from lease tract group PI4 would be brought to 

land in tankers following a route starting at PI 4 and 

continuing through T21 and T20. At least one 

transportation route must be defined for every lease 

tract group contained in an analysis, and the complete 

set of transportation routes is called the transportation scenario. The coding of program SCENARIO 

allows the inclusion of sources of oilspill risk other 

than OCS leasing in a transportation scenario.

In the preceding subsection, "Spill Occurrence Rates 

and Exposure Variables," it was explained that the exposure variable for transporting oil is the volume of oil 

handled. That is, a given volume of oil, t, moved from 

A to B can be expected to result in It spills, regardless 

of the distance between A and B. The route from A to 

B can be described as a series of launch points with the 

oilspill risks distributed among the route segments according to their length. (In figure 11, for example, 

typical weights may be 20 percent for PI4, 40 percent 

for T19, and 40 percent for T20, demonstrating a rough 

proportioning of risk to length.) Use of other exposure 

variables would require a similar weighting of transportation route segments. To accomplish this, program SCENARIO is designed with a highly flexible 

weighting process that allows the user to assign an arbitrary weight to each segment of a transportation 

route. This flexibility allows the user to specify a complicated transportation route that involves multiple 

movements of oil (e.g. "pipeline to port A, then tanker 

from port A to port B"), or to divide the oil from a lease 

tract among several different transportation routes 

(that is, "half to port A, half to port B"). If deemed justifiable, "high risk" transportation segments can even 

be assigned higher weights.

ESTIMATED VOLUMES OF OIL RESERVES

For calculating actual oilspill risks, it is necessary to 

include the volume of oil that is expected to be produced from each lease tract group as input to program 

SCENARIO. This information is compiled by the Conservation Division of the U.S. Geological Survey and is 

considered proprietary information.

PROBABILITY THAT AN OILSPILL WILL 

OCCUR AND CONTACT A TARGET

The model produces as an end result an estimated 

probability distribution for the number of spills con-
RISK CALCULATION 25

CALIFORNIA

PA CIFIC O CEAN

FIGURE 11. Potential transportation route segments for southern California, showing how oil from tract group P14 

would be brought to Long Beach via segments T20 and T 21 (Slack, Wyant, and Lanfear, 1978).

tacting each target or land segment over the production life of a lease area. This final calculation entails 

three steps which are performed by program NU: 

 For each production site or transportation route, 

the "conditional probability the probability 

that a spill, having occurred, will contact the 

given target or land segment must be extracted from the output of program HITPROB 

or program LANDSEG. (The operation of 

these programs was described in an earlier section.)

For each production site or transportation route, 

the conditional probability must be combined 

with the probability distribution of spill occur-
26 OILSPILL RISK ANALYSIS MODEL OF THE U.S. GEOLOGICAL SURVEY

rence (estimated using the methods described 

in the section "Spill Occurrence") to yield a 

single-source probability distribution for the 

number of hits on the target or land segment. 

This distribution may be arrived at by one of 

two methods, according to whether the single 

source is a "point" source, such as a production platform, or a "distributed" source, such 

as a tanker route where the oil could be released anywhere along the route.

 All single-source estimated probability distributions in a scenario (see the previous section) 

must be combined to yield the overall estimated probability distribution for the number 

of hits on each target and(or) land segment.

The following subsections describe, in more detail, the 

methods employed.

PROBABILITY OF HITS ON A TARGET 

FROM A SINGLE SOURCE

Programs HITPROB, LANDSEG, SCENARIO, 

and NU communicate through files stored on permanently mounted disk packs. After obtaining the conditional probabilities for all targets and land segments 

for each launch point, program NU begins to process 

the transportation scenario, segment by segment.

Suppose that for production platforms, the estimated probability distribution of N', the number of 

spills occurring, is negative binomial. Then, following 

from the section "Spill Occurrence,"

(4)

Suppose, further, that the conditional probability (obtained from HITPROB or LANDSEG) that a spill 

from a point source will hit a given target is p. Then, 

the estimated probability distribution of N', the 

number of spills which both occur and hit the target 

over exposure t, is negative binomial

nl(v-l)l(pt+T)n+v

with mean 

and variance

T V T /

(9)

(10)

Appendix A contains a rigorous derivation of this result.

For a distributed source defined by several transportation route segments, suppose that the estimated 

probability distribution of spills is the negative binomial with parameters as before: v, the number of past 

observed tanker spills, T, the amount of past exposure 

observed, and t, the predicted future exposure. These 

spills could occur at any point along the route. As the 

previous sections pointed out, it is often desirable in a 

risk analysis to be able to weight points along a route 

in terms of that likelihood of spill occurrence. The constraint on the weights is that the distribution of the total number of spills along the route must be the abovementioned negative binomial distribution, with mean

A=Vt/T (11)

and variance

,2 = Hi 1+_L (12)

This constraint is satisfied by assuming that the distribution of spills at each transportation route segment is 

negative binomial with parameters vz% TZ-, and fy, where 

the sum of the vz- must be v. Appendix A demonstrates 

that this structure satisfies the constraint.

To determine the estimated probability distribution 

for hits on a target from spills along the whole route, 

the model first constructs the hit distribution for each 

separate point source along the route (using the previously described methods for single point sources). 

The model then combines these distributions as described in the next section.

PROBABILITY OF HITS ON A TARGET 

FROM MULTIPLE SPILL SOURCES

The overall estimated probability distribution for 

the number of hits on a target is constructed as the 

convolution of the appropriate single-source distributions derived in previous steps. The meaning of this 

statement is best conveyed through an example: Let 

PI n be the probability of n spills hitting the target 

from the first source, and P<% n be the probability of n 

spills hitting the target from a second source,

i, n =

(13)

Let Pn represent the probability that n spills hit the 

target from both sources combined. Then,

= -*i,(H 2,o»

-* 2,1»

(14)

(15)
MODEL VERIFICATION AND LIMITATIONS 27

P* = ^1,2^2,0+^1,1^2,1 +^1,0^2,2,

and,

(16)

I ~ J.

The extension to more than two sources is similar.

Program NU carries out these calculations in the 

model. Its design is such that the effects on a risk analysis of different assumptions concerning incidence 

rates, production and transportation scenarios, or resource estimates, can be determined simply and 

straightforwardly by rerunning the program with different parameter values.

MODEL VERIFICATION 

AND LIMITATIONS

FORMAL ERROR ANALYSIS

Results of complex systems models are seldom 

amenable to formal error analysis, that is, to the expression of error in the end result as a function of errors in input quantities. Often, the error as an input 

quantity will be unobtainable or unquantifiable, or the 

error's effect on the overall analysis will be too ambiguous. Furthermore, it is especially difficult to attach 

a single number representing "standard error" to the 

results of a model run, when the results consist of a set 

of predicted probability distributions. In fact, the Bayesian methods used in constructing the distributions 

described here explicitly incorporate some elements of 

uncertainty (notably those in estimating spill incidence rate) and were developed, in part, for situations 

where classical error analysis seemed unsatisfactory.

This does not imply that a useful assessment of model reliability cannot be made, or that the results are 

categorically unreliable. Three modes of testing are 

available to the model user: (1) an informal assessment 

of individual model components is often satisfactory; 

(2) the sensitivity of the model results to particular assumptions can be tested by repeated runs with differing inputs; and (3) parts of the model can be directly 

tested by comparison with actual spill trajectories. 

The following sections contain discussions of how 

these modes of evaluation were applied to the model 

described here.

INFORMAL ERROR ANALYSIS

Several factors constrain the effective breadth of a 

model's applicability. The model's structure (how it

works, what it includes or excludes), the refinement of 

the driving data, and the analytical treatment of the 

component oilspill occurrence and movement processes all play a role. In a general purpose model, these 

limitations will differ for each application of the model. 

This requires that the assumptions necessary for specific areas be readily testable; different factors may limit 

the precision of the results from case to case. As an example, unreliable current data may not seriously affect 

model output in situations where the movement of oil 

is largely wind-dominated (as in some proposed North 

Atlantic oil production areas see Smith and others, 

1976b), but may critically affect model output where 

currents are the primary mover (as in many proposed 

Gulf of Mexico oil production areas see Wyant and 

Slack, 1978). The computer programs of the model 

have been built to facilitate case-specific testing. This 

has been done by modularizing computer programs, by 

concentrating on simple parameterizations of processes, and by restricting analytical representations of 

physical processes to those which are relatively simple, 

general, and widely accepted.

SPATIAL RESOLUTION

The model cannot represent the locations of oilspills 

or targets with any finer resolution than the cell size 

(about 1 nautical mile square) of the grid system. This 

is an artificial restriction, of course, in the sense that 

the model could be simply modified to diminish the cell 

size. Increasing the spatial resolution of the model by 

this means would, however, lead to a spurious and misleading impression of accuracy in the output, given the 

present accuracy with which the location of many targets and the spreading of large spills can be depicted.

RISK FOR NEAR-SHORE 

AND CONFINED-AREA SPILL SOURCES

The spill transport equation used in the model has 

several virtues. It is simple, is widely accepted as a reasonable representation of oilspill movement in open 

water (Stolzenbach and others, 1977, p. 5-47), and is 

void of any special assumptions which would disqualify the model for risk forecasting in most proposed offshore production areas. However, due to the fact that 

the basic oil transport equation is designed to represent the "average" movement of large spills in fairly 

open waters, the model cannot adequately represent 

the detailed movement of spills close to shore or in confined estuaries or bays, where tides and highly localized currents may dominate the movement of spilled 

oil.
28 OILSPILL RISK ANALYSIS MODEL OF THE U.S. GEOLOGICAL SURVEY

SPREADING

The model does not explicitly incorporate spreading. 

This deficiency is mitigated by several factors. First, 

because of the large regions over which the model is designed to operate and the resulting scale of model resolution, spreading is less important than overall advection in determining risks. Second, the original digitization of targets and their insertion into the grid system 

tends to expand the areas occupied by targets (any cell 

partially occupied is treated as fully occupied), and 

causes "near misses" of oilspills to be counted as hits. 

Third, recording the time-of-contact for each hit 

enables the analyst to estimate spreading effects independently, given information on oil type, sea state, and 

so on.

DECAY

The modeling of spill decay presents the same difficulties as the modeling of spreading, in that knowledge 

of factors such as oil type are integral to analytical descriptions of the physical process. As with spreading, 

the model does not explicitly calculate decay, but is 

constructed to provide information on spill travel time, 

thus enabling assessment of the extent to which decay 

might mitigate predicted impacts. Contacts of spills 

with targets are compiled in several elapsed-time categories up to 3 days, 3 to 10 days, 10 to 30 days, and 

30 to 60 days to assist the analyst in this assessment.

SENSITIVITY ANALYSIS

Sensitivity analysis of model assumptions is a useful 

technique for assessing model strengths and weaknesses. As suggested above, such analyses should be 

tailored to particular situations; different features of 

the model are critical in different situations, and the 

dictates of economy require the appropriate selection 

from the many possible sensitivity analyses. Design 

features of the model make such analyses easy to carry 

out.

Two sensitivity analyses performed during a risk analysis for proposed North Atlantic OCS production 

areas (Smith and other, 1976b), exemplify the kinds of 

analyses which can be readily performed. The basic 

transport equation includes the wind drift angle, which 

is the number of degrees wind-induced oil movements 

are deflected from the direction of the wind by Coriolis 

acceleration. Some controversy surrounds the optimal 

value of this parameter for spill modeling, but most 

suggested values fall between 0 and 20 degrees clockwise in the Northern Hemisphere (Stolzenbach and 

others, 1977, p. 81). For the North Atlantic risk analysis, two separate model runs were made using drift 

angles of 0 and 20 degrees clockwise. The resulting estimates of probabilities of spills from the proposed oil 

production areas hitting shore were 21 and 8 percent, 

respectively.

In another sensitivity analysis for the same area, actual historic wind sequences were substituted for stochastically generated ones (see the section on 

"Winds"). Table 7 shows the estimated probability of 

hitting land for spills from one proposed North Atlantic oil production site by the two modes of model operation.

These two studies convey the kinds of evaluations 

which can be conducted in the course of a risk analysis, 

and the sensitivity of results to certain key assumptions. Different sensitivities to these particular assumptions can be obtained in different OCS areas.

DIRECT MODEL VERIFICATION

Clearly, predictions of expected numbers or probabilities of spill impacts for a given place and time cannot 

be "proved" or "disproved" by a single spill. Nonetheless, a limited verification of the trajectory model was 

achieved for the area covered by the North Atlantic 

OCS oilspill risk analysis (Smith and others, 1976b). In 

December 1976 the Argo Merchant spilled 7.7 million 

gallons of oil and the spill traveled in the direction that 

the model indicated was most likely (see fig. 12). Extensive overflights and monitoring of this spill provided data for a more thorough evaluation of components of the model. In particular, by comparing actual 

and simulated spill locations, the validity of current assumptions, transport equations, and wind data source 

choices could be examined. This work, presented in detail in Grose and Mattson (1977), Pollack and Stolzenbach (1978), and Wyant and Smith (1978), supported 

the general adequacy of the transport segment of the 

model.

The spill occurred not as an idealized instantaneous 

point spill but rather was released over an extended 

period of time. To facilitate comparisons of simulated 

trajectories with the actual spill, the spill was modeled 

as a set of sequentially released points, with each 

3-hourly wind applied to the entire, gradually enlarging set. This enabled 2-dimensional construction of 

spill representations such as that in figure 12. Runs 

were made using a variety of different parameter 

values. Graphical output such as that in figure 12 

seems to be a particularly appropriate way to communicate the validity of risk forecasts to potential model 

users in that it quickly and concisely gives a feeling for 

the model's level of approximation.
MODEL OUTPUT AND CASE EXAMPLES 29

TABLE 7. Sensitivity of predicted oilspills risks for the North 

Atlantic study area to the assumption that winds can be 

modeled as a first-order Markov process.

Wind sequence Number of 

simulated 

trajectories 

per season Percent of simulated trajectories hitting shore

Winter Spring Summer Fall Total

Generated 500 1 8 19

from first

order

Markov

process.

Taken 300 0 12 8

directly

from

historic

record.

3 8

0 5

MODEL OUTPUT AND CASE EXAMPLES 

REPORTS FOR OCS LEASING

For each application of the model to a Federal OCS 

lease sale, a final report is produced which includes the 

following items:

 A discussion of the data sources which 

were used.

 Maps showing the location of the study area 

and the locations of the targets and land segments.

 Tables of conditional probabilities giving, for 

each launch point, the probabilities that an oilspill occurring at a given production site will 

contact targets or land segments within 3, 10, 

30, and 60 days.

 Tables and graphs showing the probabilities of 

oilspills occurring.

 Tables showing the overall probabilities of oilspills occurring and contacting targets or land 

segments withiri 3, 10, 30, and 60 days. 

A list of reports prepared for seven previous analyses is presented in table 8.

SUMMARY OF RESULTS TO DATE

The model has been used to conduct oilspill risk analyses for eight OCS lease sales in six Federal lease 

areas, which together represent only a small fraction of 

the total number of offshore tracts that may be developed eventually. Nevertheless, the six areas studied 

thus far are distributed among all four of the major

TABLE 8. Reports prepared for OCS lease sale analyses using the 

Oilspill Risk Analysis Model of the U.S. Geological Survey

An Oilspill Risk Analysis for the Southern California (Proposed Sale 

48) Outer Continental Shelf Lease Area; James R. Slack, 

Timothy Wyant, Kennth J. Lanfear; U.S. Geological Survey 

Water-Resources Investigations 78-80; 1978; 101 p. 

(Available from NTIS.)

An Oilspill Risk Analysis for the Mid-Atlantic (Proposed Sale 49) 

Outer Continental Shelf Lease Area; James R. Slack and 

Timothy Wyant; U.S. Geological Survey Water-Resources 

Investigation 78-56; 1978; 79 p. (Available from NTIS.)

An Oilspill Risk Analysis for the Eastern Gulf of Mexico (Proposed 

Sale 65) Outer Continental Shelf Lease Area; Timothy Wyant 

and James R. Slack, U.S. Geological Survey Open-File Report 

78-132; 1978; 72 p.

An Oilspill Risk Analysis for the Western Gulf of Alaska (Kodiak 

Island) Outer Continental Shelf Lease Area; James R. Slack, 

Richard A. Smith, and Timothy Wyant; U.S. Geological 

Survey Open-File Report 77-212;' 1977; 57 p.

An Oilspill Risk Analysis for the South Atlantic Outer Continental 

Shelf Lease Area; James R. Slack and Richard A. Smith; U.S. 

Geological Survey Open-File Report 76-653; 1976; 54 p.

An Oilspill Risk Analysis for the Mid-Atlantic Outer Continental 

Shelf Lease Area; Richard A. Smith, James R. Slack, and 

Robert K. Davis; U.S. Geological Survey Open-File Report 

76-451; 1976a; 24 p.

An Oilspill Risk Analysis for the North Atlantic Outer Continental 

Shelf Lease Area; Richard A. Smith, James R. Slack, and 

Robert K. Davis; U.S. Geological Survey Open-File Report 

76-620; 1976b; 25 p.

OCS regions which will experience oil and gas development (the Atlantic and Pacific coasts, the Gulf of Mexico, and the Alaskan Peninsula) and will serve as focal 

points for further development in those regions.

The primary objective of oilspill risk analyses conducted by the Geological Survey is to determine the 

risks of petroleum development for the tracts within a 

given lease area. Such information is useful to the Federal Government in selecting tracts to offer for sale 

from a list of tracts proposed for development by the 

oil industry. It is also of interest, however, to make 

comparisons in oilspill risk between lease areas, since 

the sites represent the four major OCS regions and the 

possibility of large differences in risk exists. An interregional comparison will be the emphasis of the summary presented here. (For more detailed descriptions 

of studies of individual lease areas, the reader is directed to the bibliography of oilspill risk reports in 

table 8.)

An important question concerning oilspill risk in 

Federal areas is whether there are significant geographic differences in spill risk per unit of expected oil 

production. (Risk per unit production can be measured 

as expected number of spill impacts on a given resource or shoreline segment per billion barrels pro-
30 OILSPILL RISK ANALYSIS MODEL OF THE U.S. GEOLOGICAL SURVEY

ATLANTIC O CEAN

Model prediction

Argo Merchant

 40° N"

Observed slick

0

I___I

50

I___I

70° W

NAUTICAL MILES

68° W

FIGURE 12. Comparison of the observed slick from the Argo Merchant with a prediction of the Oilspill Risk Analysis Model of the U.S.

Geological Survey (Wyant and Smith, 1978).

duced and transported to shore.) Differences in risk per 

unit production among sites could influence the scheduling of future lease sales. One logical policy, for example, would be to develop the sites bearing the least 

risk per unit production first, in anticipation of continual improvement in spill prevention and cleanup technology.

Table 9 gives the expected number of oilspills larger 

than 1,000 barrels occurring and reaching shore during 

the production life of the six lease areas studied. Where 

applicable, data for existing and proposed tracts are 

presented separately. Column 1 summarizes the results of trajectory model runs, and gives the range in 

conditional probability of spills reaching shore from individual production sites and transportation routes 

within each lease area, assuming a spill occurs. Column

2 gives the expected number of spill occurrences associated with both production and transportation for the 

six lease areas. Column 3 gives expected spills reaching shore during the production life of the six lease 

areas, and represents the sum of the products of conditional probabilities and expected numbers of occurrences of oilspills for individual tract groupings and 

transportation routes within each lease area. Column 4 

gives total estimated oil production for each of the six 

lease areas. Column 5 gives risk per unit production expressed as expected number of spills reaching shore 

per billion barrels of oil produced, and is calculated as 

the quotient of column 3 by column 4.

A value for the average conditional probability of 

contacting land from spill sites within a given lease 

area can be obtained by dividing the expected number
MODEL OUTPUT AND CASE EXAMPLES 31

TABLE 9. Estimated conditional probabilities and expected numbers of spills larger than 1,000 barrels reaching shore as a result of

petroleum development in each of the six Federal lease areas

Range of conditional probability of 

reaching shore from anticipated 

production areas and transportation 

Lease area routes (percent)

North Atlantic ...........

Mid-Atlantic 

Existing leases ........

South Atlantic. ..........

Eastern Gulf of Mexico

Proposed leases .......

Southern California 

Existing leases ........

Gulf of Alaska. ...........

2-79

2-42 

1-42 

69-97

17-99 

10-94

1-97 

1-97 

28-70

Expected number of 

spill occurrences

2.4

3.3 

0.86 

3.2

7.4 

0.4

9.3 

5.0 

7.5

Expected number of 

spills reaching shore

1.1

.17 

.02 

2.5

5.1 

.17

7.0 

3.8 

3.5

Estimated oil 

production 

(million 

bbls)

500

800 

150 

660

1800 

70

1330 

720 

1550

Expected number of 

spills ashore per billion 

(bbls. produced)

2.2 

0.21

0.13 

3.8

2.8 

2.5

5.3 

5.3 

2.3

of spills reaching shore (column 3) by the expected 

number of spills occurring (column 2). 1 In the North 

Atlantic, for example, the average conditional probability of a spill reaching shore, given that one has occurred on a randomly selected tract, is 46 percent 

(1.1 -j- 2.4). It can be seen from table 9, column 1, that 

even within the same lease area, the probability of oilspills reaching shore from different tracts and transportation routes is quite variable. Ranging from less 

than 20 percent to nearly 80 percent in a majority of 

lease areas, the spread in conditional probability reflects variability in wind and current patterns within 

each area as well as geographic differences, such as the 

distance of potential spill sites from shore. The variation in risk among different potential drilling sites and 

transportation routes is, in itself, evidence of the need 

for an effective methodology for estimating risk prior 

to tract selection.

More to the point of the present summary, however, 

are the large differences in oilspill risk between the 

lease areas, as seen in table 9. By far, the lowest risk of 

spills reaching shore exists in the Mid-Atlantic area, 

where the total expected number of spills reaching 

shore over the production life of both existing and proposed leases is only 0.19. In all other areas, the expectation of spills reaching shore is at least 6 times higher 

than in the Mid-Atlantic, and for southern California, 

the expectation is more than 50 times higher. A major 

reason for low risk values in the Mid-Atlantic area is 

clear in the results of trajectory model runs for that

'It should be pointed out that averages calculated in this way amount to a weighting of 

the conditional probabilities of landing from each possible spill site by the relative likelihood of spills occurring at that location.

area (Smith and others, 1976; Slack and Wyant, 1978): 

the predominance of westerly winds and the great distance of tracts from shore (50 to 100 miles) combine to 

make the conditional probability of reaching shore 

comparatively low (1 to 42 percent).

The most significant comparison of oilspill risk 

among Federal OCS areas is given in the figures for expected contacts with shoreline per unit production 

(table 9, column 5). It is worth noting that values for 

expected impacts per unit production are nearly independent of estimated oil production since production 

estimates appear both in the denominator and numerator of the calculation. Thus any errors in predicting oil 

production are not carried over into this measure of oilspill risk.

In terms of risk per unit production, the greatest 

contrast is, again, between the Mid-Atlantic lease area, 

where the expectation of spills occurring and reaching 

shore is less than one per billion barrels produced, and 

the southern California lease area, where the expectation is more than five landings per billion barrels produced. Overall, three lease areas stand out as having 

comparatively high risks of onshore impacts per unit 

production. These are the southern California, eastern 

Gulf of Mexico, and South Atlantic areas each with 

risk values greater than 2.5 landings per billion barrels 

produced. The Gulf of Alaska and the North Atlantic 

together compose a sort of medium risk category, with 

landing expectations of 2.3 and 2.2 per billion barrels, 

respectively.

All of the above statistics refer to the risk of oilspills 

reaching the shoreline within the boundaries of the digital map used to track spill trajectories. Since so many 

resources vulnerable to spilled oil are located on or
32 OILSPILL RISK ANALYSIS MODEL OF THE U.S. GEOLOGICAL SURVEY

near the shoreline, probability of contacting land is 

perhaps the best single descriptor of the risk of oilspill 

damages in OCS lease areas. However, the probability 

of contacting land is not always an indicator of the 

probability of impact on all the resources, and it is advisable to avoid condensing the description of oilspill 

risk into a single number. For this reason oilspill risk 

analyses have considered risk to an extensive list of 

specific resources (typically 20-30) for each lease area. 

Table 10 compares OCS Lease areas on the basis of 

oilspill risk to six general categories of coastal and marine resources. The second part shows the expected 

number of contacts with each resource category per 

billion barrels produced. For the most part, the oilspill 

risk values in table 10 follow the same pattern established in table 9; that is, the lowest impact probabilities appear for the Mid-Atlantic lease area, and the 

highest appear for the southern California lease area. 

There are three important instances in table 10, however, where oilspill risk is not highest for the southern 

California area. These are high-density resort and recreation areas (highest for the South Atlantic), critical 

waterfowl and seabird habitat (highest for the Gulf of 

Alaska), and marine mammal concentration areas (also 

highest for the Gulf of Alaska).

OTHER POSSIBLE USES OF THE MODEL

Although the primary purpose of the model is to assess oilspill risks from OCS lease sales, it has several 

other potential applications. Wyant and Smith (1978) 

described how the model was used in a "real time" 

mode to predict movement of oil spilled from the tanker Argo Merchant. A lease sale analysis had only recently been completed that included the area of the 

grounding, and the necessary data files were already in 

existence. Because subsequent model runs have expanded the model's data base to include major portions of the U.S. Outer Continental Shelf, operation in 

the real time mode would be possible in many other situations. Conversion to real time operation is relatively 

simple: data files must be retrieved from tape archives, 

and program SPILL must be modified so that each 

Monte Carlo trajectory run begins with a "present" 

wind velocity. However, it must be emphasized that 

such use is an extension beyond the original model design, and may not be as efficient nor as technically 

sound as using models designed specifically for oilspill 

cleanup.

The model's risk assessment capabilities are not limited to risks of OCS lease sales; other potential 

sources of oilspills, such as tanker import routes, can 

be analyzed as well. Since data files must be established for OCS lease sales in any case, the marginal 

costs of including other oilspill risks are small.

PRACTICAL ASPECTS OF OPERATING 

AND MANAGING THE MODEL

The model has been used to analyze oilspill risks in 

eight OCS lease sales, and its continued use is anticipated for future sales. To give potential users a realistic appraisal of the effort involved in model operation, 

this section discusses the practical aspects of operating the model. The management system which has 

evolved over three years of modeling operations is described, and the necessary software and hardware support for the model is identified.

MANAGEMENT SYSTEM

The model is constructed as a network of modules, or 

tasks. Each module is designed to accomplish a single 

specific objective using, as input, output produced by 

earlier modules. The major elements of a complete 

model run are illustrated in figure 13.

Modular construction is not unusual for large models, as it greatly simplifies the modification process. 

The internal workings of any module may be freely 

changed, as long as its input and output remain compatible with associated modules.

The network shown in figure 13 can produce an OCS 

lease sale analysis from data input to final report in 

four months. Initial priorities are to establish a refined 

set of input data files for program SPILL, and to identify alternative leasing and transportation scenarios. 

As program SPILL requires a substantial amount of 

computer time, every effort is made to find and correct 

errors in the data submitted to SPILL before the latter 

is executed. Trajectory test runs help to spot data errors and to identify a satisfactory set of launch points 

for potential spills. Program SPILL produces nothing 

more than a disk file containing the outcome of each 

Monte Carlo trajectory run, which subsequent programs use to generate conditional probabilities. The 

latter are combined with leasing and transportation 

scenarios to determine overall probabilities.

The different stages of model development for a typical sale may produce as many as 50 files. All of these 

are saved on disk, so that the analysis can be restarted 

at any intermediate point. Printouts associated with 

creating these files serve a valuable function in quality 

control, and help to document the progress of a model

run.

SOFTWARE

There are 21 computer programs used in the present 

version of the model, all written in IBM FORTRAN 

IV, Level H. An extensive library of subroutines and 

functions (written in either assembly language or FORTRAN), in addition to the system libraries, is also em-
PRACTICAL ASPECTS OF OPERATING AND MANAGING THE MODEL 33

TABLE 10. Expected number of oilspiU contacts with coastal and marine resources in six Federal lease areas

[*, less than 0.01; , not evaluated]

Lease area High density resort 

& recreation areas

Commercial fish, 

shellfish areas

Wildlife refuges, 

sanctuaries

Critical waterfowl, 

seabird habitat

Marine mammal 

concentrating areas

Critical habitat of

rare or endangered

species.

Contacts during production life

North Atlantic .............

Mid-Atlantic 

Existing ..............

Proposed ..............

South Atlantic .............

Eastern Gulf of Mexico 

Existing ..............

Southern California

Proposed ..............

Gulf of Alaska ............

1.0

.02

*

1.4

.92

.06

1.9

1.1

1.38

.08

.01

1.45

4.75

.24

7.0

3.6

5.5

0.43

.12

.02

.08

2.64

0.10

5.7

3.0

0.45

0.75

.15

.02

.08

.56

.02

3.00

1.45

6.62

0.1

.01

*

.03

*

*

2.66

1.43

7.5

0.17

.10

.01

.03

1.14

.08

3.91

1.97

Contacts per billion barrels of oil produced

North Atlantic .............

Mid-Atlantic

Proposed ..............

South Atlantic .............

Eastern Gulf of Mexico 

Existing ..............

Southern California

Proposed ..............

Gulf of Alaska .............

1.9

.03

*

2.1

.51

.91

1.4

1.5

2.76

.10

.07

2.18

2.63

3.53

5.3

5.0

3.5

0.86

.15

.13

.12

1.46

1.47

4.3

4 9

.29

1.5

.19

.13

.12

.31

.29

2.26

2.03

4.27

0.2

.06

*

.05

*

*

2.01

2.00

4.8

0.35

.13

.07

.05

.63

1.23

2.95

2.75

ployed. Proprietary, commercially available subroutine packages are used to control the plotting equipment.

Many of the 21 programs involve relatively straightforward processing of digitized raw data. The output 

from the present digitizing equipment used by the model requires considerable programmer intervention to 

correct both human and machine errors. In addition, 

the raw data do not always arrive in a standard format 

and often need manipulations such as map projection 

transformations. Therefore, the "front end" programs 

of the model are usually recompiled, with the necessary 

modifications, for each individual run. Complete, formal documentation is obviously difficult to achieve under these circumstances and is not expected to be completed until planned improvements in digitizing equipment are accomplished

The remaining programs of the model, including 

such major programs as WINDTRAN, SPILL, and 

NU, are stored on a disk in a partitioned data set. Their 

operation is controlled by catalogued procedures, and 

extensive checks and interlocks help to ensure correct

usage. Although design improvements still result in 

program changes, careful documentation is maintained for this group of programs.

HARDWARE

The computer hardware requirements for a model 

this size are substantial. The model is designed to be 

run on an IBM System/370, Model 155 computer. (The 

U.S. Geological Survey National Center in Reston, Virginia, has three such computer systems.) Eight hundred kilobytes (800K) of core storage is required for the 

largest program, although most of the programs need 

less than 200 kilobytes. All of the model's files are designed to be stored on a dedicated, on-line 3330 diskstorage unit; about 2,000 tracks are required for each 

OCS lease sale analysis. Tape drives, a plotter, and a 

digitizer are also required for full operation of the model. When using the model to its full capability, an analysis for an OCS lease sale can require a total of up to 

12 hours of CPU time, although no single program is 

designed to run for more than 30 minutes of CPU time.
34 OILSPILL RISK ANALYSIS MODEL OF THE U.S. GEOLOGICAL SURVEY

FIGURE 13. Flow chart illustrating the major elements of a complete model run.

Programs requiring a longer time are broken into several jobs and the output files are concatenated. Some 

programs, particularly program SPILL, are capable of

placing a noticeable strain on computer center operations, and are usually scheduled for execution at offpeak hours.
SELECTED REFERENCES 35

SELECTED REFERENCES

Alien, J., and Thanarajah, J. C. M., 1977, Laboratory studies of the 

velocity of wind on the movement of oil slicks: in Journal of 

Hydraulic Research, v. 15, no. 4.

Blummer, M., Sonza, S., and Sass, J., 1970, Hydro- carbon pollution 

of edible shellfish by an oil spill: Marine Biology International Journal on Life in Oceans and Coastal Waters, v. 5, no. 3, p. 

195-202.

Box, G. E. P., and Tiao, G. C., 1973, Bayesian inference in statistical 

analysis: Reading, Massachusetts, Addison-Wesley Publishing Company, 588 p.

Conrad, J. M., 1977, Oil spills: The policy of prevention and the 

strategy of recovery: Water Resources Research Center, University of Massachusetts, Amherst, Publication No. 93.

Csanady, G. T., 1973, Turbulent diffusion in the environment: Boston, D. Reidel Publishing Co., Geophysics and Astrophysics 

Monographs, v. 3.

Danenberger, E. P., 1976, Oil spills, 1971-75, Gulf of Mexico Outer 

Continental Shelf: U.S. Geological Survey Circular 741, 47 p.

Devanney, J. W., Ill, and Stewart, R. J., 1974, Analysis of oilspill 

statistics: Report to Council on Environmental Quality, 

Washington D.C., 126 p.

Draper, N. R., and Smith, H., 1966, Applied regression analysis: 

New York, John Wiley, 407 p.

Feller, William, 1966, An introduction to probability theory and its 

applications: New York, John Wiley, 461 p.

Grose, P. L., and Mattson, J. S. eds., 1977, The Argo Merchant 

Oilspill; A Preliminary Scientific Report: NOAA

Jeffery, P. G., 1973, Large-scale experiment on the spreading of oil 

at sea and its disappearance by natural forces, in Proceedings 

of Conference on Prevention and Control of Oil Spills (1973): 

p. 469-474.

Kirwan, A. D., Jr., McNally, G., Pazan, S., and Wert, R., 1979, 

Analysis of surface current response to wind: in Journal of 

Physical Oceanography, American Meterological Society, v. 

9, no. 2.

Murray, S. P., Smith, W. G., and Shaw, C. J., 1970, Oceanographic 

observations and theoretical analysis of oil slicks during the 

Chevron spill, March 1970: Coastal Studies Institute, Louisiana State University, Baton Rouge, Louisiana, Technical Report no. 87.

Murty, T. S., and Khandekar, M. L., 1973, Simulation of movement 

of oil slicks in the Strait of Georgia using simple atmospheric 

and ocean dynamics: in Proceedings of the 1973 Conference 

on the Prevention and Control of Oil Spills.

Nelson, W. L., 1958, Petroleum refinery engineering, New York, 

McGraw-Hill.

Offshore Oil Task Group, 1973, The Georges Bank petroleum study, 

v. II: Massachusetts Institute of Technology Sea Grant Report, 311 p.

Phillips, C. R., and Groseva, V. M., 1977, The spreading of crude 

oil spills across a lake: in Water, Air, and Soil Pollution, v. 8, 

no. 3, p. 353-360.

Pollack, A. M., and Stolzenbach, K. D., 1978, Crisis science: investigations in response to the Argo Merchant oilspill: Massachusetts Institute of Technology, Cambridge, Massachusetts.

Royer, T. R., 1979, Personal communication to D. Amstutz, U.S. 

Bureau of Land Management, March 23, 1979.

Slack, J. R., and Smith, R. A., 1976, An oilspill risk analysis for the 

South Atlantic Outer Continental Shelf lease area: U.S. Geological Survey Open-File Report 76-653, 54 p.

Slack, J. R., Smith, R. A., and Wyant, Timothy, 1977, An oilspill 

risk analysis for the Western Gulf of Alaska (Kodiak Island) 

Outer Continental Shelf lease area: U.S. Geological Survey 

Open-File Report 77-212.

Slack, J. R., and Wyant, Timothy, 1978, An oilspill risk analysis for 

the Mid-Atlantic (Proposed Sale 49) Outer Continental Shelf 

lease area: U.S. Geological Survey Open-File Report 78-56.

Slack, J. R. Wyant, Timothy, and Lanfear, K. J., 1978, An oilspill 

risk analysis for the Southern California (Proposed Sale 48) 

Outer Continental Shelf lease area: U.S. Geological Survey 

Open-File Report 78-80.

Smith, J. E., ed., 1968, Torrey Canyon pollution and marine life: 

Cambridge University Press, Cambridge, United Kingdom.

Smith, R. A., Slack, J. R., and Davis, R. K., 1976a, An oilspill risk 

analysis for the Mid-Atlantic Outer Continental Shelf lease 

area: U.S. Geological Survey Open-File Report 76-451, 24 p.

Smith, R. A., Slack, J. R., and Davis, R. K., 1976b, An oilspill risk 

analysis for the North Atlantic Outer Continental Shelf lease 

area: U.S. Geological Survey Open-File Report 76-620, 50 p.

Stewart, R. J., 1975, Oil spillage associated with the development of 

offshore petroleum resources, in Report to Organization for 

Economic Co-operation and Development, 49 p.

Stewart, R. J., 1976, A survey and critical review of U.S. oil spill 

data resources with application to the tanker/pipeline controversy: Report to U.S. Department of the Interior, Washington D.C., 69 p.

Stewart, R. J., and Kennedy, M. B., 1978, An analysis of U.S. tanker and offshore petroleum production oil spillage through 

1975: Report to Office of Policy Analysis, U.S. Department of 

the Interior, Contract Number 14-01-0001- 2193.

Stolzenbach, K.D., Madsen, O.S., Adams, E.E., Pollack, A.M., and 

Cooper, C. K., 1977, A review and evaluation of basic techniques for predicting the behavior of surface oil slicks: Ralph 

M. Parsons Laboratory, Report no. 222.

U.S. Department of the Interior, Bureau of Land Management, 

1977, Proposed 1977 Outer Continental Shelf oil and gas 

lease sale in the Western Gulf of Alaska: (OCS Draft Environmental Impact Statement), 4 volumes.

Wardley-Smith, J., ed., 1976, The control of oil pollution on the sea 

and inland waters, Graham and Trotman, Ltd., United Kingdom, 251 p.

Wyant, Timothy, and Slack, J. R., 1978, An oilspill risk analysis for 

the Eastern Gulf of Mexico (Proposed Sale 65) Outer Continental Shelf lease area: U.S. Geological Survey Open-File Report 78-132.

Wyant, Timothy, and Smith, R. A., 1978, Risk forecasting for the 

Argo Merchant spill: in Proceedings of a Symposium held 

January 11-13, 1978, at the Center for Ocean Management 

Studies, University of Rhode Island, p. 28.

Zilitinkevich, S. S., 1978, An evaluation of the oceanic surface drift 

current speed and direction: in Boundary-Layer Methodology, v. 14, no. 1.

APPENDIX
38 OILSPILL RISK ANALYSIS MODEL OF THE U.S. GEOLOGICAL SURVEY

DISTRIBUTION THEORY OF SPILL INCIDENCE

1. The derivation of the predicted probability distribution

This appendix describes rigorously the derivation of 

the predicted probability distribution on spill occurrence given as equation 4, in subsection "Predicted 

Probability Distributions for a Fixed Class of Spills." 

The development is a Bayesian one; a good general description of these Bayesian inference techniques may 

be found in Box and Tiao, (1973, p. 1-73). The application of these methods to oilspill occurrence forecasting 

was proposed and described in Devanney and Stewart, 

(1974).

We will use the following terminology: 

n= number of future spills, 

t = future exposure,

A = true rate of spill occurrence per unit exposure,

v = number of spills observed in past, 

r=past exposure,

f(n) = a marginal probability density on n, and 

f(n\y) = the conditional probability density of n 

given that the random variable y = y.

Assume that spills occur at random with some intensity, p(n):

P[n spills over exposure t] = iA£Le (A-l)

n\

Suppose that, in the absence of information about A, we 

choose to represent our uncertainty about this 

parameter in the form of an "improper" prior density 

on A:

/(A|nodata) = (A-2)

This says, in effect, that with no spills ever having 

been observed, we place a good deal of faith on A being 

equal to 0, although we allow a priori the possibility 

that it may be any positive number. This may seem artificial (as is often the case with Bayesian ignorance 

priors), but note that in any case all it takes is one observation of a spill to refute the notion that A = 0. Our 

previous feelings in the absence of any data will be 

overwhelmed by minimal experimental evidence.

Suppose we then observe v spills in T exposure and 

wish to update our estimate of A. The Bayesian approach is to represent our new estimate by a posterior 

density on A derived from our ignorance prior density 

on A combined with experimental evidence. This is accomplished through use of Bayes theorem:

. /(v|A.T)flA|no data)

f(v|A.T)/'(A|nodata) 

SQ /(v|A,T)/(A|no data)dA

/(v|A,r)/(A|no data)dA

v!

(AT)re~AT 1 dX ~^i T

(v-1)!

This is the density on A in Devanney and Stewart 

(1974, p. 28) "through which our past spill experience 

enters the analysis." It is, in Bayesian terms, the posterior density on A.

If we were to gather more evidence, this posterior 

would now become the prior, and the same reasoning 

would apply:

/(A|V2 ,T2 ) =

(A-4)

(v.+v.-l)!

Note that this is exactly the same density on A we 

would have obtained by adding the two exposures, Tlt 

and T2 , and the two numbers of spills, vl and v2, and 

treating it all as one piece of data.

Having done all this, if we desire the density of the 

phenomenon (oilspill occurrence) given our current uncertainty about A, we take the average of the Poisson 

densities weighted according to the posterior on A:

n!

n\(v-l)l(t+r)n+v

(A-5)
APPENDIX 39

This is the negative binomial distribution given as 

equation 4, in the subsection "Predicted Probability 

Distributions for a Fixed Class of Spills." 

2. Moment-generating functions

Results in the remainder of this appendix depend on 

the use of generating functions. Some standard results 

from probability theory will be reviewed.

If X is a discrete random variable with P(X = n) = 

Pn, the generating function of X (Feller, 1957, p. 249) 

is

71=0

(A-6)

Moment generating functions for some common distributions used in this analysis are as follows: 

Bernoulli random variable with probability p of "success":

(A-7)

Poisson random variable with mean At: 

= exp(A£(s-l))

Negative binomial random variable with mean vt/r and 

variance vt/r( 1 + tlr)

:( T >v (A-8)

iv is a sequence of random variables with P(X}1 = 

n) Pkn, and X is a random variable such that P(X=n) 

=pn, in order that p^n =p for any fixed n, it is necessary and sufficient that

for all s in [0, 1] (Feller, 1957, p. 262). 

If Z=X + Y, and X and Y are independent, then

(Feller, 1957, p. 250). If Xit » = 1, 2, 3,...., are independent and identically distributed,

N

= I. Xfr and N is independent of the X,% then

(A-ll)

(Feller, 1957, p. 268).

3. Convergence of the negative binomial to the 

Poisson

Let N be the number of spills in an exposure t, and 

assume (following the first part of this appendix) that 

N is a Poisson random variable with generating function

(A-12)

and that the predicted number of spills N' is a negative 

binomial random variable with generating function

(A-13)

' 't + T-fc5

where v is the number of spills observed in the past in 

the course of exposure T. If the Poisson model holds, 

then the Law of Large Numbers guarantees that as 

T-*-<» then v/T-*-A. Suppose we had adopted the negative binomial model. Then

-f

=( _v/L_)v

*t+v/A-ts' (A-14)

and, as T grows larger,

At/v + 1-^j ^ v

 v

which approaches

Ut(s-l))

(A-15)

(A-16)

as T (and hence v) grows larger.

Thus, if the Poisson model is correct, the analyst will 

be led to the Poisson model as enough data accumulates even while formally adopting the negative binomial model. Spill incidence could be modeled quite 

simply and directly using the Poisson distribution 

with A set equal to V/T. This convergence to the true 

model is an example of "Bayesian consistency." The 

advantage of the negative binomial model, as derived 

through the Bayesian methodology of this appendix, is 

that it incorporates the uncertainty about A for a finite 

exposure T, since vlt will never equal A exactly. The uncertainty is reflected in a broader distribution on spill 

incidence due to the larger variance of the negative binomial distribution a wider range of spill incident totals has non-neglible probability. The variance of N' is

 ( j, the variance of N is , and the difference

is v( )2. Thus the increase in uncertainty (as measured

by the difference in variances) is proportional to the 

squared ratio of estimated future exposure, t, to 

observed past exposure, T.
40 OILSPILL RISK ANALYSIS MODEL OF THE U.S. GEOLOGICAL SURVEY

This is only one measure of the closeness of the two 

models, of course. Of more interest is a direct comparison of the summary features presented in the Oilspill 

Risk Analysis Model of the U.S. Geological Survey, 

particularly in calculating the probability that no 

spills will occur. The expectations of N and N' are the 

same under the two models, vth. Let

spillsjPoisson

(A-18)

~P[^ spills |Negative binomial model) = 

( - )v

1 1+t/T '

Consequently, dividing the two equations and applying a Taylor's expansion yields

Pp

= + 1/2v0 2,

where

or

0<0<U/T)2

(A-19)

(A-20) 

(A-21)

Thus the difference in the probability of no spills occurring under two models (and hence the difference in the 

probabilities of one or more spills) is again directly related to the size of (t/r)2 .

4. Distribution of the total number of spills from multiple sources

Let NI and N2 be negative binomial distributed random variables with generating functions

(A-22)

2 t2S

Then,ifN=N1 +N2,

(A-23)

(A-24)

following equation A-10. In general, this will not be a 

simple distribution. However, if ti ti t and TX =TZ = 

T, then

Vi+V2,

t+f-ts

(A-25)

so N is distributed as a negative binomial random variable with mean

i=/v._i. vjl. (A-26)

and variance

(A-27)

5. Distribution of the number of hits

Let N, the total number of spills, be distributed as 

above, that is, negative binomial with parameters v, £, 

T. For each spill that occurs, associate a random variable X which takes the value 1 if a specified event occurs (such as the spill hitting land and 0 otherwise. Let 

X be a Bernoulli random variable.

1)=P (A-28)

Let T be the total number of events that occur when 

spills originate from a single source,

N 

T= S (A-29)

From section 2 of this appendix, the generating function of T is

(A-30)

From equations A-6 and A-8

-t(l-p+ps) 

T v (A-31)

Thus, the distribution of T, the number of events, is in 

turn negative binomial, but with parameters v, pt, and

T.

fi U.S. GOVERNMENT PRINTING OFFICE: 1982 S7B-O34/ 1