xploration in the United States, 1998

Reserve Estimates—

sment

ve Growth
Cover. This map represents historical oil and gas exploration and production data for the conterminous

United States and Alaska. It was derived from data used in U.S. Geological Survey Geologic

Investigations Series I-2582.* The map was compiled using Petroleum Information Corporation’s

(currently IHS Corporation) database of more than 2.2 million wells drilled in the U.S. as of June 1993.

The area of the U.S. was subdivided into 1 mi

2 grid cells for which oil and gas well completion data

were available. Each colored symbol represents a 1 mi

2 cell (to scale) for which exploration has

occurred. Each cell is identified by color as follows: red, a gas-producing cell; green, an oil-producing

cell; yellow, an oil- and gas-producing cell; gray, a cell that has been explored through drilling, but no

production has been reported. Mast and others (1998) gives details on map construction.

*Mast, R.F., Root, D.H., Williams, L.P., Beeman, W.R., and Barnett, D.L., 1998, Areas of historical oil and gas

exploration and production in the conterminous United States: U.S. Geological Survey Geologic Investigations

Series I-2582, one sheet.
Uncertainty and Inferred Reserve Estimates—

The 1995 National Assessment

By Emil D. Attanasi and Timothy C. Coburn

U.S. Geological Survey Bulletin 2172–G

U.S. Department of the Interior

U.S. Geological Survey
U.S. Department of the Interior

Gale A. Norton, Secretary

U.S. Geological Survey

Charles G. Groat, Director

U.S. Geological Survey, Reston, Virginia: 2003

This report is only available on-line at:

http://pubs.usgs.gov/bul/b2172-g/

For more information about the USGS and its products:

Telephone: 1-888-ASK-USGS

World Wide Web: http://www.usgs.gov/

Any use of trade, product, or firm names in this publication is for descriptive purposes only and does not imply 

endorsement by the U.S. Government.

Although this report is in the public domain, it contains copyrighted materials that are noted in the text. 

Permission to reproduce those items must be secured from the individual copyright owners.
iii

Contents

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

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

Acknowledgments........................................................................................................................................ 1

Concepts and Nomenclature...................................................................................................................... 1

Data................................................................................................................................................................. 3

Estimation of Inferred Reserves: Statistical Approach .......................................................................... 3

Bootstrap Procedures.................................................................................................................................. 4

Approximate Nonparametric Confidence Intervals................................................................................ 4

Summary and Implications.......................................................................................................................... 6

References Cited .......................................................................................................................................... 7

Appendix ........................................................................................................................................................ 8

Figures

1. Diagram showing layout of a discovery table for gas in gas fields................................................. 2

2. Sketch showing wells leading to additions to reserves in discovered fields................................ 2

3. Graph showing cumulative monotone field growth functions, from 1995 U.S.

Geological Survey assessment ................................................................................................. 3

4. Graphs showing base case annual additions to oil and gas reserves from oil and

gas fields discovered before 1992 ............................................................................................ 5

5. Histograms showing bootstrap distribution of inferred oil reserves and inferred

gas reserves for the 80-year period from 1992 through 2071, for fields

discovered prior to 1992 ............................................................................................................. 6

Table

1. Estimated inferred reserves, with 95 and 5 percent confidence bounds from fields

discovered as of December 31, 1991, onshore and State water areas of the

conterminous United States ...................................................................................................... 7 
Abstract

Inferred reserves are expected additions to proved 

reserves of oil and gas fields discovered as of a certain date. 

Inferred reserves accounted for 65 percent of the total oil and 

34 percent of the total gas assessed in the U.S. Geological 

Survey’s 1995 National Assessment of oil and gas in onshore 

and State offshore areas. The assessment predicted that over 

the 80-year period from 1992 through 2071, the sizes of pre1992 discoveries in the lower 48 onshore and State offshore 

areas will increase by 48 billion barrels of oil (BBO) and 313 

trillion cubic feet of wet gas (TCF). At that time, only point 

estimates were reported. This study presents a scheme to compute confidence intervals for these estimates. The recentered 

90 percent confidence interval for the estimated inferred oil of 

48 BBO is 25 BBO and 82 BBO. Similarly, the endpoints of 

the confidence interval about inferred reserve estimate of 313 

TCF are 227 TCF and 439 TCF. The range of the estimates 

provides a basis for development of scenarios for projecting 

reserve additions and ultimately oil and gas production, information important to energy policy analysis.

Introduction

Inferred reserves are expected future additions to proved 

reserves in identified fields. These resources accounted for 65 

percent of the total oil and 34 percent of the total gas assessed 

in the U.S. Geological Survey’s (USGS) 1995 National Assessment of oil and gas resources in United States onshore and 

State offshore areas (Gautier and others, 1996). The assessment 

predicted that during the 80-year period from 1992 through 

2071, sizes of pre-1992 conventional oil and gas discoveries in 

the onshore and State offshore areas of the conterminous United 

States would increase by 48 billion barrels of crude oil (BBO) 

and 313 trillion cubic feet of wet gas (TCF) (290 TCF dry gas). 

Inferred reserve estimates were calculated by the extrapolation of historical trends rather than the application of geologic 

concepts. Because the computations to derive statistical estimates of inferred reserves are complex, only point estimates 

were reported and the uncertainty of those estimates was not 

addressed. 

In recent years inferred reserves have been the dominant 

source of the Nation’s new proved reserves. Proved reserve 

levels are important because they are leading indicators of 

the sustainability of production. Because no more than 10 

to 15 percent of the proved reserves in conventional fields 

can be extracted annually without risking reservoir damage, 

proved reserves limit annual production to an amount well 

below recoverable resource volumes. Accordingly, production 

forecasts should also reflect the uncertainty associated with 

statistically based inferred reserve estimates. 

This report presents uncertainty estimates for the 1995 

inferred reserves values. The method to obtain the uncertainty estimates represents an application of the bootstrap 

theory (Efron and Tibshirani, 1998) and the algorithm used 

to calculate inferred reserves for the 1995 USGS Assessment 

(Root and others, 1997). The report is organized as follows: 

(1) data and method used for estimating inferred reserves; (2) 

the bootstrap procedures applied to compute the confidence 

intervals; (3) the implications of uncertainty in inferred reserve 

estimates to energy policy analysis. 

Ackowledgments 

The authors thank Dave Menzie, David Root, John 

Schuenemeyer, and Ione Taylor for their reviews of the methodology and the report. Lorna Carter’s editorial comments 

helped to clarify the report. 

Concepts and Nomenclature

Inferred oil and gas reserves are defined here as cumulative future additions to proved reserves in oil and gas fields 

discovered as of a certain date. Proved reserves are estimated 

quantities of hydrocarbons that geologic and engineering data 

demonstrate with reasonable certainty to be recoverable from 

identified fields under existing economic and operating conditions (Society of Petroleum Engineers, 1987; Securities and 

Exchange Commission, 1981). Inferred reserve estimates may 

specify the time period during which the additions are expected 

to occur. Here, “field growth” refers to annual additions to 

Uncertainty and Inferred Reserve Estimates—

The 1995 National Assessment

By Emil D. Attanasi and Timothy C. Coburn1

1Abilene, Texas 79699-9325.
2 Geologic, Engineering, and Assessment Studies of Reserve Growth

reserves in previously identified fields. Known recovery,

the estimate of field size used here, is defined as the sum of 

a field’s past production and its current estimate of proved 

reserves.

Analysts have frequently observed that when fields are 

grouped by year of their discovery (or vintage), the sum of 

the field size estimates (known recoveries) for each vintage 

tends to increase systematically over time. Figure 1 shows a 

partial discovery table that arrays the sum of the estimates of 

known recoveries of gas in gas fields by the particular vintage 

year shown in the first column. For each row or vintage year, 

entries from left to right represent the sum of estimates of field 

sizes (known recovery) as of 1977, 1978, ... to the last year of 

data. The systematic change across estimate years suggests that 

a statistical approach to modeling the process of field growth 

might be appropriate. 

During field development, proved reserves increase, and 

hence estimates of known recovery increase, as delineation 

drilling extends the boundary of the productive area. Estimated field size increases as the lateral boundary of the field 

is extended and as newly discovered pools that are shallower 

or deeper are added (fig. 2). Estimated field size also increases 

with improved recovery of in-place hydrocarbons resulting 

from well stimulation and fluid injection. 

Discovery No. fields 

 year 1977 1978 1979 1980 1981 1982 by vintage

1975 2,538 2,873 2,944 3,196 3,260 3,303 317

1976 1,555 1,714 1,841 2,156 2,417 2,505 340

1977 1,323 2,087 2,630 2,881 3,285 3,617 509

1978 0 1,012 1,801 2,752 2,952 3,500 422

1979 0 0 922 1,791 2,799 3,067 438

1980 000 784 1,533 2,252 453

1981 0 0 0 0 891 1,441 491

Figure 1. The layout of a discovery table: each row represents the sum of field sizes for each vintage 

(all fields with the same discovery year). Entries from left to right are successive estimates of total 

volume of gas in all gas fields in each vintage. Far right column shows the number of non-associated 

gas fields for each vintage. Source of data is the 1991 Oil and Gas Integrated Field File issued in 1993 

for fields discovered prior to 1992. (Units are in billions of cubic feet.) 

Figure 2. Schematic of wells leading to additions to reserves in discovered fields. 1, shallower pool test; 2, deeper 

pool test; 3, infill well; 4, new pool test; 5, extension or outpost (modified from Drew, 1997). In practice, the operator 

or regulatory body may classify accumulations penetrated by wells 1 through 5 as a single field or as more than one 

field. Recognition of relationships among accumulations could also be further complicated by order in which wells 

were actually drilled. Open symbol, dry hole; half-filled symbol, successful producers.

1

2 3

4

5

Known productive

limits of proven pool
The U.S. Department of Energy defines a field as an 

area consisting of a single or multiple reservoirs all grouped 

or related to the same individual geologic structure and (or) 

stratigraphic feature. Two or more reservoirs that occur in the 

field may be separated by impermeable strata vertically, laterally, or both (Energy Information Administration, 1998). In 

practice, the definition of an oil or gas field is not exact. Pools 

are not always assigned to fields for well-defined geologic 

reasons; assignments may be for regulatory convenience or 

may reflect the discovery sequence of the pools. Depending on 

the sequence of the discovery of pools, accumulations shown 

in figure 2 may be assigned to a single field or to more than 

one field. 

For any vintage, the annual growth factor for fields of age 

i is the ratio of the aggregate estimated volume of known recovery at (i+1) years after discovery to the aggregate estimated 

volume of known recovery i years after discovery. The cumulative growth factor is the ratio of the aggregate estimates of field 

sizes m years after discovery to the aggregate initial estimates 

of known recovery. Cumulative growth factors can also be 

computed as the successive products of annual growth factors. 

For each vintage, the cumulative growth function is represented 

by the series of cumulative growth factors computed for each 

successive year after discovery (fig. 3). The appendix shows 

the mathematical expressions of the growth function and the 

procedure used to compute it. 

Data

The inferred reserve estimates of the U.S. Geological 

Survey’s 1995 National Assessment were based on estimates 

of known recovery of conventional oil and gas fields discovered before 1992 and located in onshore areas and State waters 

of the United States (fields in the Federal Outer Continental Shelf are excluded). The basic data are the Oil and Gas 

Integrated Field File (OGIFF). The U.S. Energy Information 

Administration (EIA) compiles these data from proprietary 

industry responses to the “Annual Survey of Domestic Oil 

and Gas Reserves” (EIA–Form 23). The compiled data are a 

series of annual estimates, starting in 1977, of individual field 

sizes in volumes of both crude oil and wet gas. The EIA data 

consist of short series of estimates of field sizes rather than 

complete histories of fields from discovery to abandonment. 

Fields are classified as non-associated gas if the ratio of 

the estimated gas recovery to oil recovery is at least 20,000 

cubic feet of wet gas to one barrel of oil. Otherwise a field is 

assumed to be an oil field. Fields discovered before 1901 and 

those without discovery dates are excluded. Field age is the 

number of years since discovery. Production begins for most 

onshore lower 48 fields soon after discovery.

Estimation of Inferred Reserves: 

Statistical Approach

The 1995 procedure for estimation of inferred reserves 

started with the preparation of a discovery table (fig. 1) showing 

estimated volumes of gas in conventional2 gas fields by vintage 

and year of estimate. A similar table was constructed for oil 

in conventional oil fields. Cumulative growth functions (fig. 

3) were then determined using the 1993 version of the OGIFF 

(field estimates were as of the end of 1991). The method for 

determining the cumulative growth function amounted to a 

constrained optimization procedure described in detail in Attanasi and Root (1994). Two assumptions were made: on average, (1) the total known recovery of fields in specific vintage 

does not shrink as with age, and (2) the total known recovery in 

older vintages cannot grow by a larger factor in 1 year than 

total known recovery of a younger vintage. With these assumptions imposed as constraints, cumulative growth functions will 

be monotone (Appendix). After each vintage has reached 

a maximum of 91 years3 since discovery, it is no longer 

assumed to grow.

In summary, the approach to estimation of inferred 

reserves first requires that cumulative growth factors be calibrated and then used with the latest year estimate of known 

recovery for fields in each vintage to generate annual 

additions to reserves through the next 80 years (1992 to 2071). 

Specifically, growth factors and the 1991 estimates4 of known 

recovery of oil and gas fields were used to compute annual 

additions to reserves through an 80-year period starting in 

1992. The sum of annual estimates from 1992 through 2071 

constitutes the 80-year point estimate of inferred reserves. 

Uncertainty and Inferred Reserve Estimates—The 1995 National Assessment 3

Figure 3. Cumulative monotone field growth functions for oil in oil 

fields and gas in non-associated gas fields. Functions are determined using data from U.S. Energy Information Administration’s 

Oil and Gas Integrated Field File (issued in 1993 with data through 

1991). (See Root and others, 1997.)

2OGIFF data have no geologic information associated with them. Fields 

producing principally from unconventional accumulations were identified 

by geologists and excluded from the data used to determine the cumulative 

growth functions and to estimate inferred reserves.

3The 91 years represents the time span over which the growth factors were 

computed (1901–1991).

4At the time of the 1995 assessment, the 1991 estimates were the latest 

estimates of field sizes in OGIFF.

0 

2 

4 

6 

8 

10 

12 

MULTIPLE OF INITIAL ESTIMATE

0 20 40 60 80 100 

YEARS SINCE DISCOVERY

OIL IN OIL

FIELDS

GAS IN GAS

FIELDS
4 Geologic, Engineering, and Assessment Studies of Reserve Growth

The series of annual reserve additions and the inferred reserve 

estimates at 30 and 80 years are referred to here as the base 

case. Annual additions to proved reserves of (associated) 

gas from discovered oil fields were obtained by applying the 

cumulative growth function for oil to the known recovery of 

associated gas.5

Bootstrap Procedures

The bootstrap is a method for constructing the sampling 

probability distribution of sample statistic when the probability distribution governing the population is unknown. Here, 

the statistic of interest is represented by the base case estimate 

of inferred reserves (either oil or gas) for the 80-year period. 

The sampling distribution is ultimately used to construct 

confidence limits for the sample statistic (that is, the base case

estimate). The motivation for the application of the bootstrap 

is the following. 

Suppose we want to determine the sampling distribution 

of a statistic, T(x), which is calculated from observed data, 

x a vector. If T(x) is a simple analytic function of the data 

and the probability distribution Fx that generates x is known, 

a theoretical approach could be used to obtain an exact or 

approximate expression of the sampling distribution of T(x). 

If T(x) is a complicated function of the observations (as is the 

base case estimate of inferred reserves) but the probability 

distribution generating the observations is known, standard 

Monte Carlo simulation could be used to numerically generate the probability distribution for T(x). But for this case, the 

probability distribution Fx is unknown. 

The bootstrap method is based on the idea of “resampling 

with replacement” the original sample, F*, of n observations 

on which the statistic of interest, T(x), is computed. Data 

values are assumed to be equally likely events. For each 

bootstrap replicate the statistic T(x) is computed, and is now 

denoted T(x) to indicate its association with the bootstrap replicate. The procedural requirement to sample with replacement 

leads to the possibility of generating all possible combinations 

of data (Efron and Tibshirani, 1998). If the original sample 

data are such that F* (the original empirical distribution) reasonably represents Fx

 (the population), then with a sufficient 

number of bootstrap replications, the sampling distribution of 

T(x) will approximate the actual sampling distribution of T(x) 

reasonably well. 

The bootstrap procedure was applied as follows. The 

original sample is represented by the data used in the base 

case. A stratification scheme was devised so that each vintage 

(set of fields with a common discovery year) represented in 

the original discovery table comprised one of the strata. Field 

records (the series of 15 estimates, 1977 through 1991, of individual field sizes) were grouped by vintage, and the records 

for each vintage were sampled with replacement to construct 

a bootstrap replicate of size ni

, where ni is the original (total) 

number of field records for that vintage (last column of fig. 

1)6. The field records contained in each bootstrap replicate 

were summed by vintage and used to construct a bootstrap discovery table. An individual discovery table was constructed 

for each such bootstrap sample. 

To obtain the jth bootstrap estimate of inferred reserves,

Tj(x), the monotone cumulative growth function was determined using the associated bootstrap discovery table, and 

annual field growth was calculated for an 80-year period from 

1992 through 2071. To assure stability of the confidence 

intervals, 2,000 bootstrap replicate discovery tables, growth 

functions, and field growth calculations were computed for 

both oil and gas. The procedure was applied to oil and gas 

fields separately. The 2,000 bootstrap replicates for estimates,

Tj

(x), of inferred reserves for oil constituted the bootstrap distribution for oil in oil fields for the 80-year period. Similarly, 

the 2,000 bootstrap replicates of the estimates of inferred gas 

formed the bootstrap distribution for gas in gas fields for the 

80-year period. The bootstrap distributions of T(x) were the 

basis for construction of approximate 90 percent confidence 

intervals for the base case estimates of the 80-year estimates 

of inferred reserves. The 30-year inferred reserve estimates 

were also computed. 

For oil and non-associated gas, respectively, figure 4A

and figure 4B show the base case annual field growth calculations, along with the mean and median of all 2,000 bootstrap 

replicates. Parts A and B of figure 5 are the bootstrap distributions of 80-year growth (cumulative additions to reserves) for 

oil and total gas, respectively. The additions to proved reserves 

of (associated) gas from discovered oil fields are calculated 

by application of the cumulative growth function for oil to 

the known recovery of associated gas.7 The distributions 

representing 80-year inferred reserves of associated gas and 

non-associated gas were combined to compute the total gas 

distribution (fig. 5B) using Monte Carlo simulation. 

Approximate Nonparametric 

Confidence Intervals

The distributions of bootstrap estimates of inferred 

reserves were used to compute approximate confidence intervals for the base case estimates. Because the forms of the

bootstrap distributions as shown by the histograms in figure 

5A and B, were not readily identified with any known parametric distribution, a nonparametric approach to confidence 

5To preserve the distinction between the sources of additions to reserves, 

that is, oil or gas fields, the function used to calculate reserve additions of oil 

in oil fields was used to calculate associated gas reserve additions in oil fields, 

and the function used to calculate gas reserve additions in gas fields was also 

used to compute natural gas liquid reserve additions in gas fields.

6Each vintage may be visualized as an urn containing the balls corresponding to the ni

 original field records of that vintage. The sampling procedure 

was to sample each urn with replacement ni times.

7Extremely small amounts of oil in gas fields were reported in the OGIFF, 

and the changes in known recovery for this byproduct were assumed to follow 

the cumulative growth function for non-associated gas. The totals shown are 

already aggregated (probabilistically) in a fashion similar to that of the aggregation of non-associated and associated gas.
interval construction was employed. The medians of the bootstrap distributions shown in figure 5A and B deviate slightly 

from the base case8; consequently the approximate confidence 

intervals based on the bootstrap distributions were recentered 

about the base case. To accomplish this, the bias-correction 

procedure for the percentile method of confidence interval 

construction outlined by Efron and Gong (1983) was applied. 

The method proceeded as follows. 

The base case inferred reserve estimate in year k (30-year 

2021 or 80-year 2071) is defined to be pk . If the bootstrap 

distribution is centered at pk , the endpoints of a 1–2α central 

confidence interval on pk are just the α and 1ñα percentiles of 

Uncertainty and Inferred Reserve Estimates—The 1995 National Assessment 5

0 

200 

400 

600 

800 

1,000 

1,200 

1,400 

1,600 

MILLIONS OF BARRELS/YEAR

1992 2002 2012 2022 2032 2042 2052 2062 

A YEAR

0 

1,000 

2000

3,000 

4,000 

5,000 

6,000 

7,000 

BILLIONS OF CUBIC FEET/YEAR

1992 2002 2012 2022 2032 2042 2052 2062 

YEAR

Base case Bootstrap mean Bootstrap median

B

Figure 4. Base case annual estimates of additions to A, oil reserves from oil fields, 

and B, gas reserves from non-associated gas fields, discovered before 1992 in lower 

48 onshore and State water areas, along with the mean and median of the associated 

bootstrap distribution.

ˆ

ˆ

ˆ
6 Geologic, Engineering, and Assessment Studies of Reserve Growth

that distribution. When the distribution is not centered on pk , 

as in this case, a (bias) correction must be applied to the endpoints of the interval to ensure probability coverage of 1–2α.

The adjustment is based on standardizing the difference 

between pk and the median of the empirical bootstrap distribution in the following manner. Suppose Q(pk

) is the proportion 

of the empirical distribution that is smaller in value than pk . 

Let z0=S–1(Q(pk)) (5)

where S–1 is the inverse function of the standard normal 

distribution. 

The bias-corrected 1 - 2α central confidence interval on 

pk is given by 

 [Q–1(S(2z0ñzα)),Q–1(S(2z0+zα))], (6)

where zα is the 100(1–α) percentile of the standard normal 

distribution (Efron and Gong,1983), S is the standard normal 

cumulative distribution, and Q–1 is the inverse of the bootstrap distribution. If the bootstrap distribution is centered on 

pk , then Q(pk)=0.5 and z0 in equation (5) equals zero. The 

recentering of the distribution becomes more significant as the 

magnitude of the difference between pk and the median of the 

bootstrap distribution increases. 

Table 1 shows base case estimates of the cumulative 

additions to oil and gas reserves for 30 (1992–2021) and 80 

(1992–2071) years of growth (Root and others, 1997), along 

with endpoints of the associated recentered 90 percent confidence intervals. To compute the base case estimate of total 

inferred gas, the base case point estimates of non-associated 

and associated gas were simply added. However, in order 

to obtain proper confidence intervals, the bootstrap distribution of the non-associated gas inferred reserve estimates and 

distribution of the associated gas inferred reserve estimates 

were combined using Monte Carlo simulation. The bootstrap 

distributions for non-associated gas and associated gas were 

derived independently, and growth functions for gas fields and 

oil fields were determined separately. In the absence of strong 

evidence to the contrary, the aggregation assumed statistical 

independence of the non-associated gas and associated gas 

distributions. 

According to table 1 for the base case, the 30-year 

inferred oil reserve estimate represents 69 percent of the corresponding 80-year calculation. The 30-year calculation of the 

total gas inferred reserve estimate is 63 percent of the corresponding 80-year estimate. This is consistent with the profiles 

of additions to reserves shown in figure 4A and B. For the 80-

year estimates, endpoints for the recentered confidence interval for total oil are 25 BBO and 82 BBO, respectively, with 

the base case point estimate at 47 BBO. The non-associated 

gas recentered confidence interval endpoints are 153 TCF and 

356 TCF, respectively, and the base case point estimate is 223 

TCF. The endpoints of the recentered 90 percent confidence 

interval for total gas, over the 80-year period, are 227 and 439 

TCF, respectively and the base case estimate is 313 TCF (293 

dry gas). The results presented in table 1 and the distributions 

shown in figure 5 convey more information about the properties of the inferred reserve estimates than the singular point 

estimates reported in the past. 

Summary and Implications 

Note that the specific focus of this report is to provide a 

method for quantifying uncertainty that is unique to the problem of estimating inferred reserves. The estimation of inferred 

reserves departs from conventional parametric statistical methods because of the complexity of the problem and the inability 

to obtain a solution using traditional analytic methods. The 

15 35 55 75 95 115 135 155 175 195

BILLIONS OF BARRELS OF OIL

0.000

0.005

0.010

0.015

0.020

RELATIVE LIKELIHOOD

A

Figure 5. Bootstrap distribution of A, cumulative additions to oil 

reserves from fields discovered before 1992 in lower 48 onshore 

and State water areas for the 80-year period from 1992 through 

2071; B, cumulative additions to gas reserves from oil and gas 

fields discovered before 1992 in lower 48 onshore and State water 

areas for the same 80-year period as A. Smooth curves plot the 

implied density functions. 

8For total oil the base case and median are 48.1 BBO, and 52.0 BBO, and 

the total gas base case and median values are 313 and 312 TCF, respectively.

150 210 270 330 390 450 510 570 630

TRILLIONS OF CUBIC FEET OF GAS

0.000

0.002

0.004

0.006

RELATIVE LIKELIHOOD

B

ˆ

ˆ

ˆ

ˆ

ˆ ˆ

ˆ

ˆ

ˆ
Uncertainty and Inferred Reserve Estimates—The 1995 National Assessment 7

inferred estimation procedure initially relies on a form of 

constrained optimization to determine the cumulative growth 

function, where the basis of the optimization is the minimization of the sum of squared differences between estimated and 

actual values within the data. No assumptions are made about 

the underlying distribution of the differences, the presence or 

absence of correlation among them, or what they physically 

represent. The bootstrap procedure is used to obtain numerical, rather than analytical, characterization of the distribution 

of inferred reserve estimates. Finally, the bootstrap distributions of inferred reserve estimates provide the basis for 

constructing approximate confidence intervals that convey 

uncertainty about the single point estimate; and because the 

form of the bootstrap distributions cannot be established 

analytically, a nonparametric approach to confidence interval 

construction was employed. 

The individual field records spanned a 15-year period 

(1977 through 1991), which include times of boom and bust 

“typical” of the petroleum industry. The comparison of shortterm predictions with any particular near-term realization is 

not likely to be favorable because short-term proved reserve 

additions are dominated by immediate economic conditions. 

During a period of 15 years or longer, the dominant part of the 

changes in field size is related to the extension and discovery 

of new pools and the application of improved technology. 

The range of the estimates provides a rational basis for 

development of scenarios for projecting reserve additions and 

ultimately oil and gas production. Well-thought-out scenarios 

are often more appropriate than standard sensitivity analysis 

because these resource markets tend to react to change in an 

erratic fashion. Future applications of the bootstrap to the 

inferred reserve estimation problem could focus on the question of how the quality of the estimate changes as the amount 

of data, that is, years of discovery and number of years of 

estimation, increases. The uncertainty intervals could also be 

used as one of several criteria for evaluating alternative methods for estimating inferred reserves.

References Cited

Attanasi, E.D., and Root, D.H., 1994, The enigma of oil and 

gas field growth: American Association of Petroleum 

Geologists Bulletin, v. 78, no. 3, p. 321–332. 

Drew, L.J., 1997, Undiscovered petroleum and mineral 

resources—Assessment and controversy: New York, 

Plenum, 210 p. 

Efron, Bradley, and Gong, Gail, 1983, A leisurely look at the 

bootstrap, the jackknife and cross-validation: The American 

Statistician, February 1983, v. 7, no. 1, p. 36–48. 

Efron, Bradley, and Tibshirani, Robert, 1998, An introduction 

to the bootstrap: Boca Raton, Fla., Chapman and Hall/

CRC, 436 p. 

Energy Information Administration, 1998, U.S. crude oil, 

natural gas, and natural gas liquids reserves; 1997 Annual 

report: DOE/EIA-0216(97), 158 p. 

Gautier, D.L., Dolton, G.L., Takahashi, K.I., and Varnes, K.L., 

eds., 1996, 1995 National assessment of United States oil 

and gas resources—Results, methodology, and supporting 

data: U.S. Geological Survey Digital Data Series 30, version 2 corrected.1 disk.

Root, D.H., Attanasi, E.D., Mast, R.F., and Gautier, D.L., 

1997, Estimates of inferred reserves for the 1995 U.S. Geological Survey National Oil and Gas Resource Assessment: 

U.S. Geological Survey Open-File Report 95–75–L, 29 p. 

Securities and Exchange Commission, 1981, Regulation S-X 

Rule 40-10, Financial Accounting and Reporting Oil and 

Gas Producing Activities, Securities and Exchange Commission Reserves Definitions, March 1981: New York, 

Bowne and Company Inc.

Society of Petroleum Engineers, 1987, Reserves definitions

approved: Journal of Petroleum Technology, v. 39, no. 5, 

p. 576–578.

Table 1. Estimated inferred reserves, with 95 and 5 percent confidence 

bounds from fields discovered as of December 31, 1991, onshore and State 

water areas of the conterminous United States. 

Total oil (billions of barrels) 

Associated gas (trillions of cubic feet) 

Non-associated gas (trillions of cubic feet) 

Total gas (trillions of cubic feet)

 

 

 

Total oil (billions of barrels) 

Associated gas (trillions of cubic feet) 

Non-associated gas (trillions of cubic feet) 

Total gas (trillions of cubic feet) 

30-year growth

5% base 95%

19.1

29.7

102.0

151.3

33.1

197.9

138.6

59.2

193.0

102.9

52.1

254.9

5%

152.6

227.1

40.3

25.3

222.6

313.0

90.4

48.1

base

356.1

439.4

174.5

82.3

95%

80-year growth
8 Geologic, Engineering, and Assessment Studies of Reserve Growth

Appendix

For each vintage, the cumulative growth function is represented by the series of cumulative growth factors computed 

for each successive year after discovery. Changes in estimated 

known recovery conform to the relation 

ĉ(i,j+k,k)= c(i, j) [G(j+k-i)/G(j-i)] + d(i,j,k), (1)

where c(i,j) is the aggregate known recovery in year j of vintage i fields (that is, fields discovered in year i), k is elapsed 

time (in years where k>0), ĉ (i,j+k,k) is the estimate of aggregate known recovery in year j + k of vintage i fields, d(i,j,k) is 

a deviation (or error term), and G( ) is the cumulative growth 

factor.

Relation (1) implies that, for a specified vintage with 

discovery year i, the aggregate of the known field recoveries 

in year j+k can be determined by multiplying the aggregate of 

the estimated field recoveries in year j by an appropriate factor 

that is the ratio of cumulative growth factors for field ages 

(j+k-i) and (j-i). G(n) is bounded, that is G(0) = 1 and G(n) is 

a constant for n greater than 91, which is the length of the data 

series (1901–91). 

The cumulative growth factors, G(j+k-i), are computed 

using an optimization algorithm that minimizes the sum of 

squared deviations,

 ssd=∑d(i,j, k)2

, (2)

between all combinations of the actual and estimated aggregate field recoveries, ĉ (d, j+k), within the data (that is, 

discovery table) used in the analysis, where

 d(i,j,k)=c(i,j+k,k)- ĉ (i,j+k,k). (3)

The optimization is described in Attanasi and Root 

(1994). The best growth function is the set of cumulative 

growth factors that minimizes ssd.

The following condition is also imposed on the cumulative growth coefficients, 

1<G(m+1)/G(m)< G(m)/G(m-1) , m>1. (4)

The left side of the inequality ensures that vintages do 

not shrink as they age. The right side of the inequality implies 

that an older vintage cannot grow by a larger factor in one year 

than a younger vintage.

Published in the Central Region, Denver, Colorado

Manuscript approved for publication August 7, 2003

Graphics by Emil D. Attanasi 

Photocomposition by William Sowers

Edited by Lorna M.Carter