Probabilistic Methodology for 

Petroleum Resource Appraisal 

of Wilderness Lands 

By Robert A. Crovelli 

PETROLEUM POTENTIAL OF WILDERNESS LANDS IN THE 

WESTERN UNITED STATES 

GEOLOGICAL SURVEY CIRCULAR 902-0 

CONTENTS 

Page 

Abstract----------------------------------------------------~----------------- 01 

Introduction------------------------------------------------------------------- 1 

~ethodology ------------------------------------------------------------------ 2 ~arginal probability for a wilderness cluster------------------------------------- 2 

Probability distribution for a wilderness cluster----------------------------------- 2 

Lognormal distribution ofF------------------------------------------------ 2 

Lognormal distribution ofXc ----------------------------------------------- 3. 

Conditional distribution of Y c----------....,----------------------------------- 3 

Unconditional distribution ofY --------------------------------------------- 3 

Aggregations of resources in wilderness clusters ---------------------------------- 3 

Aggregations within each State--------------------------------------------- 3 

Aggregations among all States---------------------------------------------- 3 

Description of an aggregation----------------------------------------------- 3 

Dataformat---------------------------------------------------------------- 4 

Wilderness cluster data --------------------------------------------------- 4 

Provincedata------------------------------------------------------------· 4 

Example of data format---------------------------------------------------- 4 

Summary--------------------------------------------------------------------- 4 

Selected references ------------------------------------------------------------- 5 

TABLES 

Page 

TABLEl. Generalratings-------------------------------------------------------------------------------- 02 

2. Data format for assessing petroleum resources of Wilderness Lands ------------------------------------- 4 

III 
PETROLEUM POTENTIAL OF WILDERNESS LANDS IN THE WESTERN UNITED STATES 

Probabilistic Methodology for Petroleum 

Resource Appraisal of Wilderness Lands 

By Robert A. Crovelli 

ABSTRACT 

An explanation is provided of the probabilistic methodology 

developed for the quantitative analysis of the potential petroleum resources of the Wilderness Lands in the Western United 

States. This approach to the subjective probability procedures 

allows the geologists to take into account the fact that (a) petroleum resources are not evenly distributed throughout geologic 

provinces, (b) there usually has been no production of oil and 

gas in the Wilderness Lands, (c) some States containing Wilderness Lands have been explored heavily, while others have 

not been, and (d) past exploration and production results in.the 

States provide valuable information to an assessment of the 

Wilderness Lands. The outcome of all the quantitative resource 

analyses for the oil and gas potentials in the Wilderness Lands 

for each State and for the total aggregated results is described 

by using probability distributions and discussing the results 

using the range from the 95th fractile Oow estimate) to the 

5th fractile (high estimate). 

INTRODUCTION 

Probabilistic methodology was developed for a 

quantitative assessment of the petroleum potential of Wilderness Lands in the Western United 

States. The objective was to estimate, in terms of 

probability distributions, the quantity of undiscovered recoverable conventional crude oil and 

total natural gas on Wilderness Lands in each of 

the individual Western States and the total oil 

01 

and gas resources in all 11 Western States: Arizona, California, Colorado, Idaho, Montana, 

Nevada, New Mexico, Oregon, Utah, Washington, 

and Wyoming. Two probability distributions were 

derived for each of the 11 States, one for oil and 

one for total gas. Estimates of both the total oil 

and total gas resources of Wilderness Lands in all 

11 Western States were also expressed as two 

probability distributions. 

The approach to the problem that was chosen 

for this study was to utilize petroleum resource 

estimates derived from the U.S. Geological Survey 

Circular 860 study (Dolton and others, 1981; 

Varnes and others, 1982). These estimates were 

made at the geologic province level for each province and consist of: 

(a) Marginal probability or probability of 

occurrence. 

(b) Conditional probability distribution (lognormal distribution). 

(c) Unconditional probability distribution. 

A qualitative assessment of the resource potential 

of the Wilderness Lands involving classifications 

of high, medium, low, low to zero, zero, and unknown petroleum potential was also utilized and 

is discussed in chapter A in this circular by 

B. M. Miller. 
METHODOLOGY 

The basic methodology is outlined as follows: 

1. Partition all of the Wilderness Lands into geologically similar uclusters" of wilderness 

tracts. A wilderness cluster is a collection of 

one or more wilderness tracts. 

2. For each wilderness cluster within a province 

a. Determine a marginal probability for the 

cluster. 

b. Determine a conditional probability distribution for the cluster. 

c. Apply the cluster marginal probability to 

the cluster conditional probability distribution to determine the unconditional probability distribution for the cluster. 

3. Aggregate the unconditional probability distributions of resources for the clusters. 

a. Within each State. 

b. Among all States. 

MARGINAL PROBABILITY 

FOR A WILDERNESS CLUSTER 

For a wilderness cluster within a province, let 

the events be defined as 

A: Resource is present in cluster 

B: Resource is present in province 

with corresponding probabilities 

P(A): Marginal probability for cluster 

P(B): Marginal probability for province. 

P(A) and P(B) are related by 

P(A)=P(B) PCAIB) 

where 

PCAIB): Conditional probability of resource is 

present in cluster given that resource is present in province. 

There are two marginal probabilities for each 

Wilderness cluster, one for oil and one for total gas. 

PROBABILITY DISTRIBUTION 

FOR A WILDERNESS CLUSTER 

The following relationship is defined 

Yc=FXc 

where 

Y c: Conditional quantity of undiscovered 

resource in wilderness cluster 

F: Fraction of undiscovered province resource in wilderness cluster 

Xc: Conditional quantity of undiscovered 

resource in province. 

02 

The following reasonable and mathematically 

tractable assumptions are made: 

1. F is lognormally distributed. 

2. Xc is lognormally distributed. 

3. F and Xc are independent. 

Under the above assumptions, Yc is also lognormally distributed. 

LOGNORMAL DISTRIBUTION OFF 

Let 

F=faR 

where 

fa: Fraction of total province area that is the 

wilderness cluster area, i.e., the cluster 

area fraction. 

fa= Area of wilderness tract(s)/ Area of sedimentary basin or province. 

R: Rating (or richness factor) of wilderness 

cluster as to its relative potential compared to the average (mean) quantity of 

resource per unit area in the province. 

Assumptions are that fa is a fixed value because it 

is a calculated quantity, and R is a lognormal random variable where R > 0 because the lognormal 

distribution is a reasonable judgmental distribution of ratings that will make the mathematics 

tractable. These assumptions imply that F is lognormally distributed. 

The ratings approach allows the geologists to 

take into account the fact that (a) petroleum resources are not evenly distributed throughout the 

geologic provinces, (b) there usually has been no 

production of oil and gas in the Wilderness Lands, 

(c) some States containing Wilderness Lands have 

been explored heavily, while others have not 

been, and (d) past exploration and production 

results in the States would provide valuable information to an assessment of the Wilderness Lands. 

See table 1 for a general ratings scale. 

TABLE 1.-General ratings 

Qualitative assessment 

High (above average) 

Medium (average) 

Low (below average) 

Zero 

Quantitative assessment 

R>l.5 

0.5<R!S 1.5 

0<R!S0.5 

R=O 

Note: R=l assessment implies that the wilderness cluster has average potential 

compared to the province as a whole. 
Two estimates of R are made by geologists for 

each resource (two estimates for oil and two estimates for total gas): 

1. Minimum estimate of R (RmirJ 

2. Maximum estimate of R (RmaxJ 

Example: Rating of oil resource potential of 

cluster is high, 2 < R < 4. 

That is, the oil resource potential of the wilderness cluster is estimated to be between 2 and 4 

times the average oil resource potential of the 

province (per unit area). 

Hence, two estimates ofF are 

1. F min=faRmiw minimum estimate ofF 

2. F max=faRmczx, maximum estimate ofF 

The minimum and maximum estimates of F are 

used as a 99.73 percent interval estimate to determine a lognormal distribution. There are two sets 

of minimum and maximum estimates, one for oil 

and one for total gas. 

LOGNORMAL DISTRIBUTION OF Xc 

The conditional 95th and 5th fractiles of Xc are 

used to determine the corresponding lognormal 

distribution which was assumed in Dolton and 

others (1981). 

CONDITIONAL DISTRIBUTION OF Y c 

The conditional distribution of Y c is a lognormal 

distribution since the product of two independent 

lognormal random variables is lognormally 

distributed. 

UNCONDITIONAL DISTRIBUTION OF Y 

The marginal probability for the cluster, P(A), 

is multiplied times the conditional distribution of 

Y c to determine the unconditional distribution of 

Y, the unconditional quantity of undiscovered 

recoverable resource in the wilderness cluster. 

There are two probability distributions for a cluster, one for oil and one for total gas. 

AGGREGATIONS OF RESOURCES 

IN WILDERNESS CLUSTERS 

The first three unconditional central moments 

of Y are computed for each wilderness cluster. 

There is one set of three moments for oil and another set for total gas. 

03 

AGGREGATIONS WITHIN EACH STATE 

The estimates of undiscovered recoverable petroleum resources in all of the wilderness clusters in 

a State are aggregated. An assumption is made 

that the cluster estimates in a State are perfectly 

dependent, that is, fractiles are additive. This 

assumption is made because of the known geologic 

dependency of the clusters and also because the 

same geologist(s) assessed all of the clusters in a 

State. Under this assumption, the sets of three 

moments for the clusters in a State are combined 

respectively to produce the first three moments of 

the probability distribution for the State. A threeparameter lognormal distribution is fitted to the 

three moments for the State as an approximate 

aggregate distribution. 

AGGREGATIONS AMONG ALL STATES 

The estimates of undiscovered recoverable petroleum resources of the Wilderness Lands in all 11 

Western States are aggregated. There are two 

aggregations, one for oil and one for total gas. An 

assumption is made that the 11 estimates within 

an aggregation are independent. This assumption 

is made because of the wide geographic separation 

of wilderness clusters from State to State, and 

also because most of the States were assessed by 

different geologists. Since the dependency within 

a State is overstated, and the dependency between 

States is understated, the respective assumptions 

introduce compensatory effects and, furthermore, 

do not affect the means. Under the independence 

assumption, the sets of three moments for the 11 

States are combined respectively to produce the 

first three moments of the probability distribution 

for the Wilderness Lands. A three-parameter lognormal distribution is fitted to the three moments 

for the Wilderness Lands as an approximate aggregate distribution. 

DESCRIPTION OF AN AGGREGATION 

The probability distributions resulting from the 

22 State aggregations (both oil and gas) and the 

two Wilderness Lands' aggregations are described 

graphically by using the more-than (or complementary) cumulative distribution function that gives 

the probability of more than a specific amount. 

From this function, all the fractiles can be obtained. Some numerical characteristics (in million 

barrels for oil and billion cubic feet for gas) of the 

three-parameter lognormal distribution are also 
given on the graph. The mean of the probability 

distribution is used as a point estimate of the 

quantity of undiscovered recoverable resource. 

The range from the 95th fractile (the low estimate) 

to the 5th fractile (the high estimate) forms the 

reported interval estimate. 

DATA FORMAT 

The data collected depend directly upon the 

methodology and concern each wilderness cluster 

and the province containing the cluster. One set 

of data is collected for oil and one set for total gas. 

WILDERNESS CLUSTER DATA 

The wilderness cluster number, as shown on the 

data format, is an assigned code number for the 

respective wilderness cluster. The following quantity is calculated using the data from Varnes and 

Dolton (1982) and the measured cluster area: 

fa: Cluster area fraction. 

The cluster area fraction is the same for both oil and 

total gas. The following quantities are estimated: 

P(A B): Conditional probability of resource is 

present in cluster given that resource 

is present in province. 

Rmin: Minimum estimate of R, rating or 

richness factor. 

Rma.x: Maximum estimate of R, rating or 

richness factor. 

PROVINCE DATA 

The province number, as shown in the data format, is the assigned Resource Appraisal Group 

(RAG) code number of that province. The following quantities are obtained from the Circular 860 

study (Varnes and others, 1982): 

P(B): Marginal probability for province. 

F 91: Conditional 95th fractile for province. 

F 1 : Conditional 5th fractile for province. 

EXAMPLE OF DATA FORMAT 

An example of the data format, using only two 

wilderness clusters in the State of Colorado, is 

shown in table 2. 

SUMMARY 

Some special features of the developed methodology are 

1. The entire assessment area is partitioned into 

geologically similar clusters of wilderness 

tracts. 

2. Each wilderness cluster is assessed on the 

basis of its resource potential compared to 

the rest of the province. 

3. The ratings approach allows the geologists to 

take into account the fact that (a) petroleum 

resources are not evenly distributed throughout geologic provinces, (b) there has been little 

or no production of oil and gas in the Wilderness Lands, (c) some States containing Wilderness Lands have been explored heavily, 

while others have not been, and (d) past 

exploration and production results in the 

States provide valuable information to an 

assessment of the Wilderness Lands. 

4. The aggregation technique has the flexibility 

to aggregate the estimates for wilderness 

clusters assuming they are dependent within 

States and to aggregate the estimates for the 

States assuming they are independent. 

5.. Resource potential is described by using a 

probability distribution. 

TABLE 2.-Data format for assessing petroleum (oil) resources 

of Wilderness Lands in State of Colorado 

[BB, billions of barrels) 

Wilderness cluster Province 

Conditional Ratings or Conditional 

Cluster marginal Area Richness Factors RAG Marginal fractiles (BB) 

no. probability fraction province probability Low High 

P(AIB) fa Rmin Rma.x no. P(B) Fell F~~ 

1 1.0 0.00124 2 4 85 1.0 0.24 3.22 

2 0.8 0.00215 0.5 1 85 1.0 0.24 3.22 

04 
6. A point estimate using the mean of the distribution is computed along with an interval 

estimate using the range from the 95th fractile (the low estimate) to the 5th fractile (the 

high estimate). 

SELECTED REFERENCES 

Crovelli, R. A., 1981, Probabilistic methodology for oil and 

gas resource appraisal: U.S. Geological Survey Open-File 

Report 81-1151, 77 p. 

Dolton, G. L., Carlson, K. H., Charpentier, R. R., Coury, A. B., 

Crovelli, R. A., Frezon, S. E., Khan, A. S., Lister, J. H., 

McMullin, R. H., Pike, R. S., Powers, R. B., Scott, E. W., 

and Varnes, K. L., 1981, Estimates of undiscovered recoverable conventional resources of oil and gas in the United 

States: U.S. Geological Survey Circular 860, 87 p. 

05 

Ford, C. E., McLaren, R. A., 1980, Methods for obtaining 

distributions of uranium occurrence from estimates of geologic features: Oak Ridge, Tenn., K/CSD-13, Union Carbide, Computer Services Division, 121 p., 1 microfiche. 

Miller, B. M., Thomsen, H. L., Dolton, G. L., Coury, A. B., 

Hendricks, T. A., Lennartz, F. E., Powers, R. B., Sable, E. G., 

and Varnes, K. L., 1975, Geological estimates of undiscovered recoverable oil and gas resources in the United States: 

U.S. Geological Survey Circular 725, 78 p. 

Varnes, K. L., Charpentier, R. R., and Dolton, G. L., 1982, 

Conditional estimates and marginal probabilities for undiscovered recoverable oil and gas resources by provincestatistical background data for U.S. Geological Survey Circular 860: U.S. Geological Survey Open-File Report 82-

666-A, 29 p. 

Varnes, K. L., and Dolton, G. L., 1982, Estimated areas and 

volumes of sedimentary rock in the United States by 

province-statistical background data for U.S. Geological 

Survey Circular 860: U.S. Geological Survey Open-File 

Report 82-666-C, 11 p.