Typing Mineral Deposits Using Their Grades and Tonnages in an Artificial Neural Network

A test of the ability of a probabilistic neural network to classify deposits into types on the basis of deposit tonnage and average Cu, Mo, Ag, Au, Zn, and Pb grades is conducted. The purpose is to examine whether this type of system might serve as a basis for integrating geoscience information available in large mineral databases to classify sites by deposit type. Benefits of proper classification of many sites in large regions are relatively rapid identification of terranes permissive for deposit types and recognition of specific sites perhaps worthy of exploring further.Total tonnages and average grades of 1,137 well-explored deposits identified in published grade and tonnage models representing 13 deposit types were used to train and test the network. Tonnages were transformed by logarithms and grades by square roots to reduce effects of skewness. All values were scaled by subtracting the variable's mean and dividing by its standard deviation. Half of the deposits were selected randomly to be used in training the probabilistic neural network and the other half were used for independent testing. Tests were performed with a probabilistic neural network employing a Gaussian kernel and separate sigma weights for each class (type) and each variable (grade or tonnage).Deposit types were selected to challenge the neural network. For many types, tonnages or average grades are significantly different from other types, but individual deposits may plot in the grade and tonnage space of more than one type. Porphyry Cu, porphyry Cu-Au, and porphyry Cu-Mo types have similar tonnages and relatively small differences in grades. Redbed Cu deposits typically have tonnages that could be confused with porphyry Cu deposits, also contain Cu and, in some situations, Ag. Cyprus and kuroko massive sulfide types have about the same tonnages. Cu, Zn, Ag, and Au grades. Polymetallic vein, sedimentary exhalative Zn-Pb, and Zn-Pb skarn types contain many of the same metals. Sediment-hosted Au, Comstock Au-Ag, and low-sulfide Au-quartz vein types are principally Au deposits with differing amounts of Ag.Given the intent to test the neural network under the most difficult conditions, an overall 75% agreement between the experts and the neural network is considered excellent. Among the largestclassification errors are skarn Zn-Pb and Cyprus massive sulfide deposits classed by the neuralnetwork as kuroko massive sulfides—24 and 63% error respectively. Other large errors are the classification of 92% of porphyry Cu-Mo as porphyry Cu deposits. Most of the larger classification errors involve 25 or fewer training deposits, suggesting that some errors might be the result of small sample size. About 91% of the gold deposit types were classed properly and 98% of porphyry Cu deposits were classes as some type of porphyry Cu deposit. An experienced economic geologist would not make many of the classification errors that were made by the neural network because the geologic settings of deposits would be used to reduce errors. In a separate test, the probabilistic neural network correctly classed 93% of 336 deposits in eight deposit types when trained with presence or absence of 58 minerals and six generalized rock types. The overall success rate of the probabilistic neural network when trained on tonnage and average grades would probably be more than 90% with additional information on the presence of a few rock types.     

Uncertainty Estimate in Resources Assessment: A Geostatistical Contribution

For many decades the mining industry regarded resources/reserves estimation and classification as a mere calculation requiring basic mathematical and geological knowledge. Most methods were based on geometrical procedures and spatial data distribution. Therefore, uncertainty associated with tonnages and grades either were ignored or mishandled, although various mining codes require a measure of confidence in the values reported. Traditional methods fail in reporting the level of confidence in the quantities and grades. Conversely, kriging is known to provide the best estimate and its associated variance. Among kriging methods, Ordinary Kriging (OK) probably is the most widely used one for mineral resource/reserve estimation, mainly because of its robustness and its facility in uncertainty assessment by using the kriging variance. It also is known that OK variance is unable to recognize local data variability, an important issue when heterogeneous mineral deposits with higher and poorer grade zones are being evaluated. Altenatively, stochastic simulation are used to build local or global uncertainty about a geological attribute respecting its statistical moments. This study investigates methods capable of incorporating uncertainty to the estimates of resources and reserves via OK and sequential gaussian and sequential indicator simulation The results showed that for the type of mineralization studied all methods classified the tonnages similarly. The methods are illustrated using an exploration drill hole data sets from a large Brazilian coal deposit.     

Monte Carlo Simulations of Product Distributions and Contained Metal Estimates

Estimation of product distributions of two factors was simulated by conventional Monte Carlo techniques using factor distributions that were independent (uncorrelated). Several simulations using uniform distributions of factors show that the product distribution has a central peak approximately centered at the product of the medians of the factor distributions. Factor distributions that are peaked, such as Gaussian (normal) produce an even more peaked product distribution. Piecewise analytic solutions can be obtained for independent factor distributions and yield insight into the properties of the product distribution. As an example, porphyry copper grades and tonnages are now available in at least one public database and their distributions were analyzed. Although both grade and tonnage can be approximated with lognormal distributions, they are not exactly fit by them. The grade shows some nonlinear correlation with tonnage for the published database. Sampling by deposit from available databases of grade, tonnage, and geological details of each deposit specifies both grade and tonnage for that deposit. Any correlation between grade and tonnage is then preserved and the observed distribution of grades and tonnages can be used with no assumption of distribution form.     

Information structure analysis for quantitative assessment of mineral resources and the discovery of a silver deposit

We propose an information-structure-analysis (ISA) method to quantify the correlations between quantitative and qualitative variables as well as within each type of variable. This method is applied to the evaluation of mineral resources in the western Zheijiang Province of China. The district contains a number of silver-bearing Fe–Cu–Pb–Zn mineral deposits near igneous complexes and FeCuPbZn zones away from the complexes. Silver anomalies occur not only in the known Fe–Cu–Zn–Pb deposits, but also in the country rock, suggesting the possible existence of silver deposits far from the igneous complexes.The tonnage distribution of silver is modeled by Monte Carlo simulation. This simulation is conducted on the basis of the correlations between silver (Ag) and lead (Pb), since no known data on silver is available. The known tonnage distribution of lead in 11 control cells was used to approximate the tonnage distribution of silver in the Monte Carlo simulation. With ISA and Monte Carlo methods, the total amount of potential polymetallic resources in 49 cells in the western Zhejiang Provice is predicted. Significantly, a deposit with about 24 tonnes of silver has been found within our exploration target area.     

Basic concepts in three-part quantitative assessments of undiscovered mineral resources

Since 1975, mineral resource assessments have been made for over 27 areas covering 5×10⁶ km² at various scales using what is now called the three-part form of quantitative assessment. In these assessments, (1) areas are delineated according to the types of deposits permitted by the geology,(2) the amount of metal and some ore characteristics are estimated using grade and tonnage models, and (3) the number of undiscovered deposits of each type is estimated.Permissive boundaries are drawn for one or more deposit types such that the probability of a deposit lying outside the boundary is negligible, that is, less than 1 in 100,000 to 1,000,000.     

Map Scale Effects on Estimating the Number of Undiscovered Mineral Deposits

Estimates of numbers of undiscovered mineral deposits, fundamental to assessing mineral resources, are affected by map scale. Where consistently defined deposits of a particular type are estimated, spatial and frequency distributions of deposits are linked in that some frequency distributions can be generated by processes randomly in space whereas others are generated by processes suggesting clustering in space. Possible spatial distributions of mineral deposits and their related frequency distributions are affected by map scale and associated inclusions of non-permissive or covered geological settings. More generalized map scales are more likely to cause inclusion of geologic settings that are not really permissive for the deposit type, or that include unreported cover over permissive areas, resulting in the appearance of deposit clustering. Thus, overly generalized map scales can cause deposits to appear clustered. We propose a model that captures the effects of map scale and the related inclusion of non-permissive geologic settings on numbers of deposits estimates, the zero-inflated Poisson distribution. Effects of map scale as represented by the zero-inflated Poisson distribution suggest that the appearance of deposit clustering should diminish as mapping becomes more detailed because the number of inflated zeros would decrease with more detailed maps. Based on observed worldwide relationships between map scale and areas permissive for deposit types, mapping at a scale with twice the detail should cut permissive area size of a porphyry copper tract to 29% and a volcanic-hosted massive sulfide tract to 50% of their original sizes. Thus some direct benefits of mapping an area at a more detailed scale are indicated by significant reductions in areas permissive for deposit types, increased deposit density and, as a consequence, reduced uncertainty in the estimate of number of undiscovered deposits. Exploration enterprises benefit from reduced areas requiring detailed and expensive exploration, and land-use planners benefit from reduced areas of concern.     

Geological Modeling of Gold Deposit Based on Grade Domaining Using Plurigaussian Simulation Technique

Mineral resource evaluation requires defining grade domains of an ore deposit. Common practice in mineral resource estimation consists of partitioning the ore body into several grade domains before the geostatistical modeling and estimation at unsampled locations. Many ore deposits are made up of different mineralogical ensembles such as oxide and sulfide zone: being able to model the spatial layout of the different grades is vital to good mine planning and management. This study addresses the application of the plurigaussian simulation to Sivas (Turkey) gold deposits for constructing grade domain models that reproduce the contacts between different grade domains in accordance with geologist’s interpretation. The method is based on the relationship between indicator variables from grade distributions on the Gaussian random functions chosen to represent them. Geological knowledge is incorporated into the model by the definition of the indicator variables, their truncation strategy, and the grade domain proportions. The advantages of the plurigaussian simulation are exhibited through the case study. The results indicated that the processes are seen to respect reproducing complex geometrical grades of an ore deposit by means of simulating several grade domains with different spatial structure and taking into account their global proportions. The proposed proportion model proves as simple to use in resource estimation, to account for spatial variations of the grade characteristics and their distribution across the studied area, and for the uncertainty in the grade domain proportions. The simulated models can also be incorporated into mine planning and scheduling.     

Economic Filters Applied to Quantitative Mineral Resource Simulations

The U.S. Geological Survey has developed a technique that allows mineral resource experts to apply economic filters to estimates of undiscovered mineral resources. This technique builds on previous work that developed quantitative methods for mineral resource assessments. A Monte-Carlo calculation uses mineral deposit models to estimate commodity grades and tonnages of undiscovered deposits. The results then are analyzed using simple estimates of capital expenditures and daily operating costs for a mine and associated mill. The daily operating costs and the value of the ore are used to calculate the net present value of the deposit, which is compared to the capital expenditures to determine whether the deposit is economic. Repetition of these calculations for many deposits produces a table that can be interpreted in terms of the probability of there being deposits that have anet present value exceeding some specified amount. Sample calculations indicate that applying economic filters to simulated mineral resources might change the perception of the results compared to presenting the calculations in terms of the expected mean gross-in-place value of the minerals.     

Probabilistic Estimates of Number of Undiscovered Deposits and Their Total Tonnages in Permissive Tracts Using Deposit Densities

Empirical evidence indicates that processes affecting number and quantity of resources in geologic settings are very general across deposit types. Sizes of permissive tracts that geologically could contain the deposits are excellent predictors of numbers of deposits. In addition, total ore tonnage of mineral deposits of a particular type in a tract is proportional to the type’s median tonnage in a tract. Regressions using size of permissive tracts and median tonnage allow estimation of number of deposits and of total tonnage of mineralization. These powerful estimators, based on 10 different deposit types from 109 permissive worldwide control tracts, generalize across deposit types. Estimates of number of deposits and of total tonnage of mineral deposits are made by regressing permissive area, and mean (in logs) tons in deposits of the type, against number of deposits and total tonnage of deposits in the tract for the 50th percentile estimates. The regression equations (R ² = 0.91 and 0.95) can be used for all deposit types just by inserting logarithmic values of permissive area in square kilometers, and mean tons in deposits in millions of metric tons. The regression equations provide estimates at the 50th percentile, and other equations are provided for 90% confidence limits for lower estimates and 10% confidence limits for upper estimates of number of deposits and total tonnage. Equations for these percentile estimates along with expected value estimates are presented here along with comparisons with independent expert estimates. Also provided are the equations for correcting for the known well-explored deposits in a tract. These deposit-density models require internally consistent grade and tonnage models and delineations for arriving at unbiased estimates.     

Using PETRIMES to estimate mercury deposits in California

In this article, we examine the use of an unconventional procedure, PETRIMES, to estimate mineral resources of mercury deposits in California. The study, which is based on the nonparametric discovery process model and Q-Q plots, suggests that a lognormal distribution is appropriate for the mercury deposits in California. The results of the assessment are summarized as follows: (1) the total number of mercury deposits in the population is approximately 165; (2) the median value of the largest undiscovered deposit size is 487 flasks; (3) the mean of the remaining mercury potential is 2,500 flasks; and (4) the population resource ranges from 1,040,000 to 4,300,000 flasks (at a 0.9 probability level).     

Metallic Mineral Resources in the Twenty-First Century: II. Constraints on Future Supply

Supplying worldwide demand of metallic raw materials throughout the rest of this century may require 5–10 times the amount of metals contained in known ore deposits. This demand can be met only if mineral deposits containing the required masses of metals, in excess of present day ore reserves, exist in the Earth’s crust. It is, by definition, not known whether or not such mineral deposits exist. On the basis of the statistical distribution of metal tonnages contained in known ore deposits, however, it is possible to place constraints on the size distribution of the deposits that must be discovered in order to meet the expected demand. A nondimensional analysis of the distribution of metal tonnages in deposits of 20 metals shows that most of them follow distributions that, although not strictly lognormal, share important characteristics with a lognormal distribution. Chief among these is the observation that frequency falls off symmetrically and geometrically with deposit size, relative to a median deposit size that is approximately equal to the geometric mean deposit size. An immediate consequence of this behavior is that most of the metal endowment is concentrated in deposits that are several orders of magnitude larger than the median deposit size, and that are much rarer than the most common deposits that cluster around the median deposit size. The analysis reveals remarkable similarities among the statistical distributions of most of the metals included in this study, in particular, the fact that distribution of most metals can be fully described with essentially the same value (about 2–3) of the scale parameter, σ, which is the only parameter needed to describe the behavior of a normalized lognormal variable. This observation makes it possible to derive the following general conclusions, which are applicable to most metals—both scarce and abundant. First, it is unlikely that undiscovered mineral deposits of sizes comparable to those that contain most of the known metal endowment exist in sufficient quantities to supply the expected worldwide demand throughout the rest of this century. Second, if the expected demand is to be met, one must hope that very large deposits, perhaps up to one order of magnitude larger than the largest known deposits, exist in accessible portions of the Earth’s crust, and that these deposits are discovered.     

A Chart for Judging Optimal Sample Spacing for Ore Grade Estimation

Mineral deposit grades are usually estimated using data from samples of rock cores extracted from drill holes. Commonly, mineral deposit grade estimates are required for each block to be mined. Every estimated grade has always a corresponding error when compared against real grades of blocks. The error depends on various factors, among which the most important is the number of correlated samples used for estimation. Samples may be collected on a regular sampling grid and, as the spacing between samples decreases, the error of grade estimated from the data generally decreases. Sampling can be expensive. The maximum distance between samples that provides an acceptable error of grade estimate is useful for deciding how many samples are adequate. The error also depends on the geometry of a block, as lower errors would be expected when estimating the grade of large-volume blocks, and on the variability of the data within the region of the blocks. Local variability is measured in this study using the coefficient of variation (CV). We show charts analyzing error in block grade estimates as a function of sampling grid (obtained by geostatistical simulation), for various block dimensions (volumes) and for a given CV interval. These charts show results for two different attributes (Au and Ni) of two different deposits. The results show that similar errors were found for the two deposits, although they share similar features: sampling grid, block volume, CV, and continuity model. Consequently, the error for other attributes with similar features could be obtained from a single chart.     

The study of the undiscovered mineral resources of the Tongass National Forest and adjacent lands,Southeastern Alaska

The quantitative probabilistic assessment of the undiscovered mineral resources of the 17.1-million-acre Tongass National Forest (the largest in the United States) and its adjacent lands is a nonaggregated, mineral-resource-tract-oriented assessment designed for land-planning purposes. As such, it includes the renewed use of gross-in-place values (GIPV's) in dollars of the estimated amounts of metal contained in the undiscovered resources as a measure for land-use planning.Southeastern Alaska is geologically complex and contains a wide variety of known mineral deposits, some of which have produced important amounts of metals during the past 100 years. Regional geological, economic geological, geochemical, geophysical, and mineral exploration history information for the region was integrated to define 124 tracts likely to contain undiscovered mineral resources. Some tracts were judged to contain more than one type of mineral deposit. Each type of deposit may contain one or more metallic elements of economic interest. For tracts where information was sufficient, the minimum number of as-yet-undiscovered deposits of each type was estimated at probability levels of 0.95, 0.90, 0.50, 0.10, and 0.05.The undiscovered mineral resources of the individual tracts were estimated using the U.S. Geological Survey's MARK3 mineral-resource endowment simulator; those estimates were used to calculate GIPV's for the individual tracts. Those GIPV's were aggregated to estimate the value of the undiscovered mineral resources of southeastern Alaska. The aggregated GIPV of the estimates is $40.9 billion.Analysis of this study indicates that (1) there is only a crude positive correlation between the size of individual tracts and their mean GIPV's: and (2) the number of mineral-deposit types in a tract does not dominate the GIPV's of the tracts, but the inferred presence of synorogenic-synvolcanic nickel-copper, porphyry copper skarn-related, iron skarn, and porphyry copper-molybdenum deposits does. The influence of this study on the U.S. Forest Service planning process is yet to be determined.     

Estimation of undiscovered deposits in quantitative mineral resource assessments—examples from Venezuela and Puerto Rico

Quantitative mineral resource assessments used by the United States Geological Survey are based on deposit models. These assessments consist of three parts: (1) selecting appropriate deposit models and delineating on maps areas permissive for each type of deposit; (2) constructing a grade-tonnage model for each deposit model; and (3) estimating the number of undiscovered deposits of each type. In this article, I focus on the estimation of undiscovered deposits using two methods: the deposit density method and the target counting method.In the deposit density method, estimates are made by analogy with well-explored areas that are geologically similar to the study area and that contain a known density of deposits per unit area. The deposit density method is useful for regions where there is little or no data. This method was used to estimate undiscovered low-sulfide gold-quartz vein deposits in Venezuela.Estimates can also be made by counting targets such as mineral occurrences, geophysical or geochemical anomalies, or exploration plays and by assigning to each target a probability that it represents an undiscovered deposit that is a member of the grade-tonnage distribution. This method is useful in areas where detailed geological, geophysical, geochemical, and mineral occurrence data exist. Using this method, porphyry copper-gold deposits were estimated in Puerto Rico.     

Quantitative assessment of future development of cooper/silver resources in the Kootenai National Forest,Idaho/Montana: Part I—Estimation of the copper and silver endowments

Faced with an ever-increasing diversity of demand for the use of public lands, managers and planners are turning more often to a multiple-use approach to meet those demands. This approach requires the uses to be mutually compatible and to utilize the more valuable attributes or resource values of the land. Therefore, it is imperative that planners be provided with all available information on attribute and resource values in a timely fashion and in a format that facilitates a comparative evaluation.The Kootenai National Forest administration enlisted the U.S. Geological Survey and U.S. Bureau of Mines to perform a quantitative assessment of future copper/silver production potential within the forest from sediment-hosted copper deposits in the Revett Formation that are similar to those being mined at the Troy Mine near Spar Lake. The U.S. Geological Survey employed a quantitative assessment technique that compared the favorable host terrane in the Kootenai area with worldwide examples of known sediment-hosted copper deposits. The assessment produced probabilistic estimates of the number of undiscovered deposits that may be present in the area and of the copper and silver endowment that might be contained in them.Results of the assessment suggest that the copper/silver deposit potential is highest in the southwestern one-third of the forest. In this area there is an estimated 50 percent probability of at least 50 additional deposits occurring mostly within approximately 260,000 acres where the Revett Formation is thought to be present in the subsurface at depths of less than 1,500 meters. A Monte Carlo type simulation using data on the grade and tonnage characteristics of other known silver-rich, sediment-hosted copper deposits predicts a 50 percent probability that these undiscovered deposits will contain at least 19 million tonnes of copper and 100,000 tonnes of silver. Combined with endowments estimated for identified, but not thoroughly explored deposits, and deposits that might also occur in the remaining area of the forest, the endowment potential increases to 23 million tonnes of copper and 190,000 tonnes of silver.     

Examining Risk in Mineral Exploration

Successful mineral exploration strategy requires identification of some of the risk sources and considering them in the decision-making process so that controllable risk can be reduced. Risk is defined as chance of failure or loss. Exploration is an economic activity involving risk and uncertainty, so risk also must be defined in an economic context. Risk reduction can be addressed in three fundamental ways: (1) increasing the number of examinations; (2) increasing success probabilities; and (3) changing success probabilities per test by learning. These provide the framework for examining exploration risk. First, the number of prospects examined is increased, such as by joint venturing, thereby reducing chance of gambler's ruin. Second, success probability is increased by exploring for deposit types more likely to be economic, such as those with a high proportion of world-class deposits. For example, in looking for 100+ ton (>3 million oz) Au deposits, porphyry Cu-Au, or epithermal quartz alunite Au types require examining fewer deposits than Comstock epithermal vein and most other deposit types. For porphyry copper exploration, a strong positive relationship between area of sulfide minerals and deposits' contained Cu can be used to reduce exploration risk by only examining large sulfide systems. In some situations, success probabilities can be increased by examining certain geologic environments. Only 8% of kuroko massive sulfide deposits are world class, but success chances can be increased to about 15% by looking in settings containing sediments and rhyolitic rocks. It is possible to reduce risk of loss during mining by sequentially developing and expanding a mine—thus reducing capital exposed at early stages and reducing present value of risked capital. Because this strategy is easier to apply in some deposit types than in others, the strategy can affect deposit types sought. Third, risk is reduced by using prior information and by changing the independence of trials assumption, that is, by learning. Bayes' formula is used to change the probability of existence of the deposit sought on the basis of successive exploration stages. Perhaps the most important way to reduce exploration risk is to employ personnel with the appropriate experience and yet who are learning.     

Developing a geographic expert system for regional mapping of volcanogenic massive sulfide (VMS) deposit potential

A personal computer-based geographic information system (GIS) is used to develop a geographic expert system (GES) for mapping and evaluating volcanogenic massive sulfide (VMS) deposit potential. The GES consists of an inference network to represent expert knowledge, and a GIS to handle the spatial analysis and mapping. Evidence from input maps is propagated through the inference network, combining information by means of fuzzy logic and Bayesian updating to yield new maps showing evaluation of hypotheses. Maps of evidence and hypotheses are defined on a probability scale between 0 and 1. Evaluation of the final hypothesis results in a mineral potential map, and the various intermediate hypotheses can also be shown in map form.The inference net, with associated parameters for weighting evidence, is based on a VMS deposit model for the Chisel Lake deposit, a producing mine in the Early Protoerzoic Snow Lake greenstone belt of northwest Manitoba. The model is applied to a small area mapped at a scale of 1:15,840. The geological map, showing lithological and alteration units, provides the basic input to the model. Spatial proximity to contacts of various kinds are particularly important. Three types of evidence are considered: stratigraphic, heat source, and alteration. The final product is a map showing the relative favorability for VMS deposits. The model is implemented as aFortran program, interfaced with the GIS. The sensitivity of the model to changes in the parameters is evaluated by comparing predicted areas of elevated potential with the spatial distribution of known VMS occurrences.     

模型多参数灵敏度与不确定性分析

以潮白河为研究区域,探讨了与模型参数及模型模拟性能有关的多参数灵敏度及不确定性分析方法(Multi-Parameter Sensitivity and Uncertainty Analysis, MPSUA)。基于Monte Carlo模拟的多参数灵敏度分析,可以评价模型中多个参数的相对重要性。GLUE不确定性分析则能对模型性能进行量化评估。实例研究表明,通过MPSUA方法,可以减少优化参数的个数。而且,在没有对模型进行参数优化之前,基于MPSUA就可以确定模型的模拟精度。例如同样的模型在潮河可以获得比在白河更高的模拟精度。这种同一模型在不同流域所体现的差异性,一方面是源于模型结构本身的不完善,另一方面则与用于建模的数据误差有关。SCE-UA参数优化结果与MPSUA结果几乎一致,说明本文的参数灵敏度与模型总体性能评估方法比较合理。     

Conditional estimates of the number of podiform chromite deposits

A desirable guide for estimating the number of undiscovered mineral deposits is the number of known deposits per unit area from another well-explored permissive terrain. An analysis of the distribution of 805 podiform chromite deposits among ultramafic rocks in 12 subareas of Oregon and 27 counties of California is used to examine and extend this guide. The average number of deposits in this sample of 39 areas is 0.225 deposits per km² of ultramafic rock; the frequency distribution is significantly skewed to the right. Probabilistic estimates can be made by using the observation that the lognormal distribution fits the distribution of deposits per unit area. A further improvement in the estimates is available by using the relationship between the area of ultramafic rock and the number of deposits.The number (N) of exposed podiform chromite deposits can be estimated by the following relationship: log₁₀(N)=–0.194+0.577 log₁₀(area of ultramafic rock). The slope is significantly different from both 0.0 and 1.0. Because the slope is less than 1.0, the ratio of deposits to area of permissive rock is a biased estimator when the area of ultramafic rock is different from the median 93 km². Unbiased estimates of the number of podiform chromite deposits can be made with the regression equation and 80 percent confidence limits presented herein.