Robert A. Crovelli

An objective replacement method for censored geochemical data

Geochemical data are commonly censored, that is, concentrations for some samples are reported as “less than” or “greater than” some value. Censored data hampers statistical analysis because certain computational techniques used in statistical analysis require a complete set of uncensored data. We show that the simple substitution method for creating an uncensored dataset, e.g., replacement by3/4 times the detection limit, has serious flaws, and we present an objective method to determine the replacement value. Our basic premise is that the replacement value should equal the mean of the actual values represented by the qualified data. We adapt the maximum likelihood approach (Cohen, 1961) to estimate this mean. This method reproduces the mean and skewness as well or better than a simple substitution method using3/4 of the lower detection limit or3/4 of the upper detection limit. For a small proportion of “less than” substitutions, a simple-substitution replacement factor of 0.55 is preferable to3/4; for a small proportion of “greater than” substitutions, a simple-substitution replacement factor of 1.7 is preferable to4/3, provided the resulting replacement value does not exceed 100%. For more than 10% replacement, a mean empirical factor may be used. However, empirically determined simple-substitution replacement factors usually vary among different data sets and are less reliable with more replacements. Therefore, a maximum likelihood method is superior in general. Theoretical and empirical analyses show that true replacement factors for “less thans” decrease in magnitude with more replacements and larger standard deviation; those for “greater thans” increase in magnitude with more replacements and larger standard deviation. In contrast to any simple substitution method, the maximum likelihood method reproduces these variations. Using the maximum likelihood method for replacing “less thans” in our sample data set, correlation coefficients were reasonably accurately estimated in 90% of the cases for as much as 40% replacement and in 60% of the cases for 80% replacement. These results suggest that censored data can be utilized more than is commonly realized.

Probabilistic Assessment of Precipitation-Triggered Landslides Using Historical Records of Landslide Occurrence, Seattle, Washington

Ninety years of historical landslide records were used as input to the Poisson and binomial probability models. Results from these models show that, for precipitation-triggered landslides, approximately 9 percent of the area of Seattle has annual exceedance probabilities of 1 percent or greater. Application of the Poisson model for estimating the future occurrence of individual landslides results in a worst-case scenario map, with a maximum annual exceedance probability of 25 percent on a hillslope near Duwamish Head in West Seattle. Application of the binomial model for estimating the future occurrence of a year with one or more landslides results in a map with a maximum annual exceedance probability of 17 percent (also near Duwamish Head). Slope and geology both play a role in localizing the occurrence of landslides in Seattle. A positive correlation exists between slope and mean exceedance probability, with probability tending to increase as slope increases. Sixty-four percent of all historical landslide locations are within 150 m (500 ft, horizontal distance) of the Esperance Sand/Lawton Clay contact, but within this zone, no positive or negative correlation exists between exceedance probability and distance to the contact.

FASP,an analytic resource appraisal program for petroleum play analysis

Abstract An analytic probabilistic methodology for resource appraisal of undiscovered oil and gas resources in play analysis is presented in a FORTRAN program termed FASP. This play-analysis methodology is a geostochastic system for petroleum resource appraisal in explored as well as frontier areas. An established geologic model considers both the uncertainty of the presence of the assessed hydrocarbon and its amount if present. The program FASP produces resource estimates of crude oil, nonassociated gas, dissolved gas, and gas for a geologic play in terms of probability distributions. The analytic method is based upon conditional probability theory and many laws of expectation and variance.

The generalized 20/80 law using probabilistic fractals applied to petroleum field size

Fractal properties of the Pareto probability distribution are used to generalize “the 20/80 law.” The 20/80 law is a heuristic law that has evolved over the years into the following rule of thumb for many populations: 20 percent of the population accounts for 80 percent of the total value. The generalp100/q100 law in probabilistic form is defined withq as a function ofp, wherep is the population proportion andq is the proportion of total value. Using the Pareto distribution, thep100/q100 law in fractal form is derived with the parameterq being a fractal, whereq unexpectedly possesses the scale invariance property. The 20/80 law is a special case of thep100/q100 law in fractal form.Thep100/q100 law in fractal form is applied to petroleum fieldsize data to obtainp andq such thatp100% of the oil fields greater than any specified scale or size in a geologic play account forq100% of the total oil of the fields. The theoretical percentages of total resources of oil using the fractalq are extremely close to the empirical percentages from the data using the statisticq. Also, the empirical scale invariance property of the statisticq for the petroleum fieldsize data is in excellent agreement with the theoretical scale invariance property of the fractalq.

Probabilistic methodology for estimation of undiscovered petroleum resources in play analysis of the United States

A geostochastic system called FASPF was developed by the U.S. Geological Survey for their 1989 assessment of undiscovered petroleum resources in the United States. FASPF is a fast appraisal system for petroleum play analysis using a field-size geological model and an analytic probabilistic methodology. The geological model is a particular type of probability model whereby the volumes of oil and gas accumulations are modeled as statistical distributions in the form of probability histograms, and the risk structure is bilevel (play and accumulation) in terms of conditional probability. The probabilistic methodology is an analytic method derived from probability theory rather than Monte Carlo simulation. The resource estimates of crude oil and natural gas are calculated and expressed in terms of probability distributions. The probabilistic methodology developed by the author is explained.The analytic system resulted in a probabilistic methodology for play analysis, subplay analysis, economic analysis, and aggregation analysis. Subplay analysis included the estimation of petroleum resources on non-Federal offshore areas. Economic analysis involved the truncation of the field size with a minimum economic cutoff value. Aggregation analysis was needed to aggregate individual play and subplay estimates of oil and gas, respectively, at the provincial, regional, and national levels.

U.S. Geological Survey Assessment Methodology for Estimation of Undiscovered Petroleum Resources in Play Analysis of the Arctic National Wildlife Refuge

The objective of this investigation was to develop a geostochastic system for estimation of undiscovered petroleum resources in play analysis. A resource appraisal system was designed for play analysis using a reservoir-engineering model and an analytic probabilistic methodology. The system resulted in a package of computer programs, called FASP, for play analysis, subplay analysis, mixed-lithology analysis, and aggregations. Resource estimates of crude oil, nonassociated gas, dissolved gas, and total gas are calculated in terms of probability distributions as tables and graphs. The FASP computer package is a large complex system that is very flexible and efficient, but which can easily be adapted to most microcomputers. This system was used by the U.S. Geological Survey to assess the in-place petroleum potential of the Arctic National Wildlife Refuge located in northeastern Alaska.

U.S. Geological Survey probabilistic methodology for oil and gas resource appraisal of the United States

Probabilistic methodology used by the U.S. Geological Survey is described for estimating the quantity of undiscovered recoverable conventional resources of oil and gas in the United States. A judgmental probability distribution of the “quantity of resource” and its properties is determined for a geologic province or basin. From this distribution, point and interval estimates of the quantity of undiscovered resource are obtained. Distributions and their properties are established for each of the following resources: (1) oil and nonassociated gas from estimates of the probability of the resource being present and the conditional probability distribution of the quantity of resource given that the resource is present, (2) associated-dissolved gas from its corresponding oil distribution, (3) total gas, (4) oil and total gas in two or more provinces. Computer graphics routines are illustrated with examples from the U.S. Geological Survey Circular 860.

A simplified spreadsheet program for estimating future growth of oil and gas reserves

Reserve growth refers to the typical increases in estimated sizes of fields that occur through time as oil and gas fields are developed and produced. Projections of the future reserve growth of known fields have become important components of hydrocarbon resource assessments. In this paper, we present an algorithm for estimating the future reserve growth of known fields. The algorithm, which incorporates fundamental reserve-growth assumptions used by others in the past, is programmed for a personal computer in the form of formulas for a spreadsheet. The primary advantages of this spreadsheet program lie in its simplicity and ease of use. We also present a library of 17 different growth functions that provides numerical models for predicting the future sizes of existing oil and gas fields in various regions of the United States. These growth functions are formatted for use in the spreadsheet program.

Comparative study of aggregations under different dependency assumptions for assessment of undiscovered recoverable oil resources in the world

The U.S. Geological Survey assessed all significant sedimentary basins in the world for undiscovered conventionally recoverable crude-oil resources. Probabilistic methodology was applied to each basin assessment to produce estimates in the form of probability distributions. Basin probability distributions were computer aggregated to produce resource estimates for the entire world. Aggregation was approximated by a three-parameter lognormal distribution by combining the first three central moments of basin distributions. For purposes of experiment and study, world aggregation was conducted under four different sets of assumptions. The four cases are (1) dependent assessments of all basins, (2) dependent assessments within continental areas, but independent assessments among continental areas, (3) dependent assessments within countries, but independent assessments among countries, and (4) independent assessments of all basins. Mean estimate remained the same in all four cases, but the width of interval estimate formed using the 95th and 5th fractiles decreased with reduced dependency in going from first to fourth case.

U.S. department of the interior U.S. geological survey: Fractal lognormal percentage analysis of the U.S. geological survey’s 1995 national assessment of conventional oil and gas resources

The U.S. Geological Survey periodically makes appraisals of the oil and gas resources of the Nation. In its 1995 National Assessment the onshore areas and adjoining State waters of the Nation were assessed. As part of the 1995 National Assessment, 274 conventional oil plays and 239 conventional nonassociated-gas plays were assessed. The two datasets of estimates studied herein are the following: (1) the mean, undiscovered, technically recoverable oil resources estimated for each of the 274 conventional oil plays, and (2) the mean, undiscovered, technically recoverable gas resources estimated for each of the 239 conventional nonassociatedgas plays. It was found that the two populations of petroleum estimates are both distributed approximately as lognormal distributions. Fractal lognormal percentage theory is developed and applied to the two populations of petroleum estimates. In both cases the theoretical percentages of total resources using the lognormal distribution are extremely close to the empirical percentages from the oil and nonassociated-gas data. For example, 20% of the 274 oil plays account for 73.05% of the total oil resources of the plays if the lognormal distribution is used, or for 75.52% if the data is used; 20% of the 239 nonassociated-gas plays account for 76.32% of the total nonassociated-gas resources of the plays if the lognormal distribution is used, or for 78.87% if the data is used