Handbook of Mathematical Geosciences pp 375393  Cite as
Advances in Sensitivity Analysis of Uncertainty to Changes in Sampling Density When Modeling Spatially Correlated Attributes
Abstract
A comparative analysis of distance methods, kriging and stochastic simulation is conducted for evaluating their capabilities for predicting fluctuations in uncertainty due to changes in spatially correlated samples. It is concluded that distance methods lack the most basic capabilities to assess reliability despite their wide acceptance. In contrast, kriging and stochastic simulation offer significant improvements by considering probabilistic formulations that provide a basis on which uncertainty can be estimated in a way consistent with practices widely accepted in risk analysis. Additionally, using real thickness data of a coal bed, it is confirmed once more that stochastic simulation outperforms kriging.
19.1 Introduction
In any form of sampling, there is always significant interest in establishing the reliability that may be placed on any conclusions extracted from a sample of certain size. In the earth sciences and engineering, such conclusions can be the extension of a contamination plume or the in situ resources of a mineral commodity. Increases in sample size result in monotonic improvements with diminishing returns: up to measuring the entire population, the benefits increase with the number of observations. In the classical statistics of independent random variables, the number of observations is all that counts. In spatial statistics, however, the locations of the data are also important.
Early on in spatial sampling, it was recognized that sampling distance was a factor in determining the reliability of estimations. However, insurmountable difficulties of incorporating other factors led to the reliability of spatial samplings being determined solely by geographical distance, particularly for the public disclosure of mineral resources (e.g., USBM and USGS 1976).
Significant advances in the determination of spatial uncertainty did not take place until the advent of digital computers and the formulation of geostatistics (e.g., Matheron 1965). Geostatistics introduced the concept of kriging variance, which was a significant improvement over the relatively simplistic distance criteria for determining reliability. The third generation of methods to determine reliability of spatial sampling came with the development of spatial stochastic simulation shortly after the formulation of kriging (Journel 1974).
Although there are several reports in the literature about applications of distance methods (e.g., USGS 1980; Wood et al. 1983; Rendu 2006) and kriging (e.g., Olea 1984; Bhat et al. 2015), the mere fact that distance methods are still being used indicates that the merits of the geostatistical methods remain unappreciated. This chapter is an application of the three families of methods for conducting sensitivity analyses on the reliability of the assessment of geologic resources due to variations in sample spacing. The simulation formulation given here is novel as it is an illustrative example used for comparing all three approaches.
19.2 Data
Thickness data. ID = identification number; Thick. = thickness; ft = feet
ID  Easting (ft)  Northing (ft)  Thick. (ft)  ID  Easting (ft)  Northing (ft)  Thick. (ft) 

2  431,326  1,316,298  49.0  39  399,741  1,236,607  77.0 
3  398,753  1,316,124  32.0  40  432,107  1,236,582  70.5 
4  352,156  1,316,015  37.0  41  384,280  1,236,527  58.0 
5  365,531  1.315,818  49.0  42  415,737  1,236,459  78.0 
7  382,816  1,314,601  55.0  44  352,743  1,221,026  10.0 
9  430,850  1,301,568  48.0  45  368,483  1,220,742  26.0 
10  398,805  1,301,506  57.0  46  431,473  1,220,645  59.0 
11  352,234  1,299,533  37.0  47  399,596  1,220,598  92.0 
12  366,769  1,300,871  50.0  48  415,871  1,220,477  86.0 
13  414,876  1,300,240  56.0  49  384,411  1,220,477  32.0 
14  382,892  1,299,775  58.0  51  367,180  1,206,180  17.0 
16  416,097  1,284,247  60.0  52  399,353  1,205,960  99.0 
17  430,593  1,284,243  47.0  53  417,304  1,204,922  76.0 
18  400,291  1,284,132  87.0  54  384,456  1,204,470  28.0 
19  384,138  1,283,859  53.0  55  432,027  1,203,507  52.0 
20  368,123  1,283,849  56.0  56  351,466  1,203,245  11.0 
21  351,956  1,283,728  36.0  123  356,115  1,295,788  35.0 
23  366,138  1,268,773  55.0  145  360,095  1,291,759  38.0 
24  383,559  1,268,661  60.0  166  362,980  1,289,047  42.0 
25  431,915  1,268,363  70.0  216  371,863  1,277,272  50.0 
26  415,962  1,268,347  75.0  234  377,019  1,272,660  57.0 
27  399,884  1,268,270  63.0  282  387,755  1,264,534  60.0 
28  352,933  1,268,254  34.0  299  391,477  1,261,727  70.0 
30  352,738  1,253,951  21.0  318  395,814  1,257,798  58.0 
31  384,499  1,253,969  62.0  380  403,832  1,248,290  75.0 
32  400,076  1,252,554  79.0  406  407,848  1,243,143  81.0 
33  415,868  1,256,420  57.0  427  411,790  1,240,470  84.0 
34  368,579  1,250,159  44.0  470  419,690  1,232,449  92.0 
35  430,979  1,251,072  81.5  497  422,447  1,228,465  90.0 
37  352,979  1,237 155  30.0  512  427,604  1,224,598  46.0 
38  368,493  1,236,862  37.0  1001  415,000  1,316,000  45.0 
With resources of more than 200 billion short tons of coal in place, the Gillette coal field is one of the largest coal deposits in the United States (Luppens et al. 2008). There are eleven beds of importance in the field. The Anderson coal bed, in the Paleocene Tongue River Member of the Fort Union Formation, is the thickest and most laterally continuous of the six most economically significant beds. This low sulfur, subbituminous coal has a field average thickness of 45 ft. Hence, it is the main mining target.
19.3 Traditional Uncertainty Assessment

0 to ¼ mi: measured

¼ to ¾ mi: indicated

¾ to 3 mi: inferred

More than 3 mi: hypothetical

5–10 ft, probability 0.3

10–15 ft, probability 0.4

15–21 ft, probability 0.2

21–28 ft, probability 0.1
Note that the sum of the probabilities of all possible outcomes is 1.0. Random variables rigorously allow answering multiple questions about unknown magnitudes, in this case, the likely thickness to penetrate. A sample of just three assertions would be: (a) coal will certainly be intersected because the value zero is not listed among the possibilities; (b) it is more likely that the intersected thickness will be less than 15 ft than greater than 15 ft; and (c) odds are 6 to 4 that the thickness will be between 10 and 21 ft, or to put it differently, the 11 ft interval between 10 and 21 ft has a probability of 0.6 of containing the true thickness. These are the standard concepts and tools used universally in statistics to characterize uncertainty.

The classification uses an ordinal scale (e.g., Urdan 2017), supposedly ranked, but the classification does not indicate how much more uncertain one category is relative to another. In practice, it has been found that errors may not be significantly different among categories (Olea et al. 2011).

The results of a distance classification are difficult to validate. The tonnage in a class denotes an accumulated magnitude over an extensive volume of the deposit. The entire portion of the deposit comprising a class would have to be mined in order to determine the exact margin of error in the classification for such a class. In practical terms, the classification is not falsifiable, thus it is unscientific (Popper 2000). Moreover, there is little value in determining the reliability of a prediction post mining.

The classification fails to consider the effect of geologic complexity. Coal deposits ordinarily contain several geologically different beds that may be penetrated drilling a single hole. When all beds are penetrated by the same vertical drill holes, the drilling pattern is the same for all beds. Using the Circular 891 classification method, the areal extension of each category is the same for the resources of each coal bed separately and for the accumulated resources considering all coal beds, while logic indicates that the extension of true reliability classes should be all different.

For similar reasons, in a multiseam deposit, increasing the drilling density results in the same reduction in uncertainty for all coalbeds, which is also unrealistic.

The number of methods for estimating resources is continuously growing, hopefully for the better. Considering that not all methods are equally powerful, independently of the data, different methods offer varying degrees of reliability. The uncertainty denoted by the Circular 891 classification is insensitive to the methods used in the calculation of the tonnage. For example, inferred resources remain as inferred resources independently of the nature and quality of the methods used in the assessment.
Despite these drawbacks and the formulation of the superior alternatives below, Circular 891 and similar approaches remain the prevailing methods worldwide for the public disclosure of uncertainty in the assessment of mineral resources and reserves (JORC 2012; CRIRSCO 2013).
19.4 Kriging
Kriging is a family of spatial statistics methods formulated for the improvement in the reporting of uncertainty and in the estimation of the attributes of interest themselves. Although it is possible to establish links between kriging and other older estimation methods in various disciplines, mining was the driving force behind the initial developments of kriging and other related methods collectively known today as geostatistics (Cressie 1990).
 \( n \le N \)

is a subset of the sample consisting of the observations closest to \( {\mathbf{s}}_{o} \);
 \( \gamma ({\mathbf{d}}) \)

is the semivariogram, a function of the distance d between two locations;
 \( \lambda_{i} \)

is a weight determined by solving a system of linear equations comprising semivariogram terms; and
 \( \mu \)

is a Lagrange multiplier, also determined by solving the same system of equations.
The method presumes knowledge of the function characterizing the spatial correlation between any two points, which is never the case. A structural analysis must be conducted before running kriging to estimate this function: a covariance or semivariogram. The semivariogram can be regarded as a scaled distance function. The weights and the Lagrange multiplier depend on the semivariogram for multiple drillhole to drillhole distances and estimation location to drillhole distances. For details, see for example Olea (1999).
The two terms, \( z^{*} ({\mathbf{s}}_{o} ) \) and \( \sigma^{2} ({\mathbf{s}}_{o} ) \), are the mean and the variance of the random variable modeling the uncertainty of the true value of the attribute \( z({\mathbf{s}}_{o} ) \), terms that are compatible with all that is known about the attribute through the sample of size N. Variance is a measure of dispersion, in this case, dispersion of possible values around the estimate, which is the most likely value. Hence, changing the sample, a sensitivity analysis of kriging variance is a sensitivity analysis of variations in uncertainty due to changes in the sampling scheme. From Eq. 19.2, the kriging variance does not depend directly on the observations. The dependence is only indirect through the semivariogram, which is based on the data. Considering that there is one true semivariogram per attribute, changes in adequate sampling should not result in significant changes in the estimated semivariogram, which is kept constant. This independence between data and standard error facilitates the application of kriging to the sensitivity analysis in the reliability of an assessment due to changes in sampling strategy because mathematically actual measurements are not necessary to calculate standard errors; the modeler only has to specify the semivariogram and the sampling locations.
Kriging is able to provide random variables for the statistical characterization of uncertainty if the modeler is willing to introduce a distributional assumption. \( z^{*} ({\mathbf{s}}_{o} ) \) and \( \sigma^{2} ({\mathbf{s}}_{o} ) \) are the mean and the variance of the distribution of the random variable providing the likely values for \( z({\mathbf{s}}_{o} ) \). These parameters are necessary but not sufficient to fully characterize any distribution. However, this indetermination can be eliminated by assuming a distribution that is fully determined with these two parameters. Ordinarily, the distribution of choice is the normal distribution, followed by the lognormal. The form of the distribution does not change by subtracting \( z({\mathbf{s}}_{o} ) \) from all estimates. As the difference \( z^{*} ({\mathbf{s}}_{o} )  z({\mathbf{s}}_{o} ) \) is the estimation error, the distributional assumption also allows characterizing the distribution for the error at \( {\mathbf{s}}_{o} \).

It is possible to calculate the probability that the true value of the attribute lies in any number of intervals. Probabilities are a form of a ratio variable, for which zero denotes an impossible event and, say, a 0.2 probability denotes twice the likelihood of occurrence of an event than 0.1.

Validation is modular. An adequate theory assures that, on average, \( z^{*} ({\mathbf{s}}_{o} ) \) and \( \sigma^{2} ({\mathbf{s}}_{o} ) \) are good estimates of reality. Yet, as illustrated by an example in the last Section, if going ahead with validation of the uncertainty modeling primarily to check the adequacy of the normality assumption, it is not necessary to validate all possible locations throughout the entire deposit to evaluate the quality of the modeling.

The effect of complexity in the geology is taken into account by the semivariogram.

In general, the thickness of every coal bed or the accumulated values of thickness for several coal beds has a different semivariogram. Thus, even if the sampling configuration is the same, the standard error maps will be different.

The characterization of uncertainty is specific to the estimation method because the results are valid only for estimated values using the same form of kriging used to generate the standard errors.
19.5 Stochastic Simulation
Despite limited acceptance, the kriging variance has been in use for a while in the sensitivity analysis of uncertainty to changes in sampling distances and configurations (e.g., Olea 1984; Cressie et al. 1990). Kriging, like any mathematical method, has been open to improvements. One result has been the formulation of another family of methods: stochastic simulation.
Relative to the topic of this chapter, stochastic simulation offers two improvements: (a) it is no longer necessary to assume the form for the distribution providing all possible values for the true value of the attribute \( z({\mathbf{s}}_{o} ) \); and (b) the standard error is sensitive to the data.
19.6 Validation
19.7 Conclusions
Distance methods, kriging and stochastic simulation rank, in that order, in terms of increasing detail and precision of the information that they are able to provide concerning the uncertainty associated to any spatial resource assessment.
The resource classification provided by distance methods is completely independent of the geology of the deposit and the method applied to calculate the mineral resources. The magnitude of the resource per class has no associated quantitative measure of the deviation that could be expected between the calculated resource and the actual amount in place.
The geostatistical methods of kriging and stochastic simulation base the modeling on the concept of random variable used in statistics, which allows the same type of probabilistic forecasting used in other forms of risk assessments. Censored data were used for validating the accuracy of the probabilistic predictions that can be made using the geostatistical methods. The results were entirely satisfactory, particularly in the case of stochastic simulation.
Notes
Acknowledgements
This contribution completed a required review and approval process by the U.S. Geological Survey (USGS) described in Fundamental Science Practices (http://pubs.usgs.gov/circ/1367/) before final inclusion in this volume. I wish to thank Brian Shaffer and James Luppens (USGS), Peter Dowd (University of Adelaide) and Josep Antoni MartínFernández (Visiting Fulbright Scholar, USGS) for suggestions leading to improvements to earlier versions of the manuscript.
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