Natural Resources Research

, Volume 21, Issue 4, pp 443–459

Sequential Simulation Approach to Modeling of Multi-seam Coal Deposits with an Application to the Assessment of a Louisiana Lignite

Authors

    • U.S. Geological Survey
  • James A. Luppens
    • U.S. Geological Survey
Article

DOI: 10.1007/s11053-012-9185-1

Cite this article as:
Olea, R.A. & Luppens, J.A. Nat Resour Res (2012) 21: 443. doi:10.1007/s11053-012-9185-1

Abstract

There are multiple ways to characterize uncertainty in the assessment of coal resources, but not all of them are equally satisfactory. Increasingly, the tendency is toward borrowing from the statistical tools developed in the last 50 years for the quantitative assessment of other mineral commodities. Here, we briefly review the most recent of such methods and formulate a procedure for the systematic assessment of multi-seam coal deposits taking into account several geological factors, such as fluctuations in thickness, erosion, oxidation, and bed boundaries. A lignite deposit explored in three stages is used for validating models based on comparing a first set of drill holes against data from infill and development drilling. Results were fully consistent with reality, providing a variety of maps, histograms, and scatterplots characterizing the deposit and associated uncertainty in the assessments. The geostatistical approach was particularly informative in providing a probability distribution modeling deposit wide uncertainty about total resources and a cumulative distribution of coal tonnage as a function of local uncertainty.

Keywords

Krigingsequential simulationprobability distributioncoal bedassessment

Introduction

Mining resource assessments in general and coal resource assessments, in particular, are characterized by fluctuations in the attributes of interest that are difficult to infer from the incomplete information typically available ahead of mining, a situation that invariably leads to uncertain results (e.g., Caers 2011). Olea et al. (2011) presented a methodology for a single bed that was more mathematically satisfactory and yielded better quantitative results than conventional uncertainty approaches that simply evaluate proximity to the nearest coal measurement, such as those proposed by Wood et al. (1983). Also, the new methodology expresses uncertainty about total resources in terms of in-place tonnage, rather than distance units. This article builds on the 2011 article by assessing a multi-bed deposit to continue exploring the promising potential of geostatistical methods that have been recognized progressively as the standard approach to mineral estimation since the middle of the past century (Mwasinga 2001). In the last two decades, the popularity of geostatistical methods has shifted away from kriging to spatial stochastic simulation (e.g., Srivastava 1994; Journel and Kyriakidis 2004). The following authors have applied stochastic simulation to the particular case of coal assessments:

Hohn and McDowell (2001) (a) modeled three coal beds in properties of different areal extensions at two 7½ min quadrangles in southern West Virginia and (b) reported the resources of each seam separately.

de Souza et al. (2004) (a) applied the modeling to a single layer deposit and (b) assessed a portion of the deposit delimited by the lease rather than by the natural boundary of the coal bed.

Heriawan and Koike (2008) (a) used the spatial modeling method OPSIM for interpolation of discontinuous bodies (Koike and Matsuda 2005) and (b) did not aggregate the resources.

It may be relevant to note that none of these authors checked the results against an exhaustive sampling or some form of ground truthing. In addition, no account was taken for oxidized coal of no commercial value.

Our approach uses three different geostatistical methods and data from a multi-bed lignite deposit in Louisiana to practically:
  1. (a)

    determine the outer areal boundary amenable for modeling;

     
  2. (b)

    combine surface elevation and model of depth of oxidation to establish the upper boundary of minable resources in a deposit;

     
  3. (c)

    make simultaneous use of drilling information in terms of holes penetrating and missing seams for estimating missing coal areas within the model boundary;

     
  4. (d)

    model coal bed thickness and related uncertainty away from drill holes;

     
  5. (e)

    prepare a probabilistic assessment of total resources aggregating three beds using an average coal density;

     
  6. (f)

    use a function of expected cell tonnage as a model for local uncertainty.

     

The measurements are supposed to be free of institutional uncertainties of non-geologic nature, such as insufficient depth, improper sampling, or inconsistencies in the correlations.

This article is an expansion of the modeling of a single-bed deposit (Olea et al. 2011). The main objective is testing the capability of geostatistics to properly assess uncertainty for the more complex case of multiple coal beds.

Methodology

Spatial modeling methods in general and geostatistics, in particular, do not work well far beyond peripheral observations inferring what some authors denote as hypothetical resources. Hence, a first step in any geostatistical modeling is the definition of the area feasible to model using the data available. In the absence of legal or geological constraints, one possibility is to limit the study area employing some criteria based on distance to drill holes. A related but more optimal approach is to use the kriging variance. Kriging can be regarded as a generalization of the least squares method to take into account spatial correlation among the data (e.g., Isaaks and Srivastava 1989). The method infers values one location at a time, generating an estimate and its error variance—also known as the kriging variance. This variance is sensitive to the distance and pattern of all measurements around any estimation location as well as to the style and magnitude of the fluctuations according to the data. Thus, by defining a threshold, one can use the kriging variance to define the extension of the area amenable of modeling. Given a number of drill holes, the kriging variance is consistent with the experience that more continuous depositional systems, such as a lacustrine deposit, are easier to assess and extrapolate than more complex deposits, such as deltaic deposits.

In the early days of geostatistics, modeling was based on a straight application of kriging (e.g., Tewalt et al. 1983). Today such practice has been abandoned primarily for three reasons. First, locally, estimates are conditionally biased, i.e., estimated values toward the low end are systematically higher than the actual values and the converse is true for the highest values. Hence, the probability distribution and spatial autocorrelation of the estimated attribute do not follow the probability distribution and the autocorrelation of the data, let alone that of the true attribute. Second, if there is interest in assigning confidence intervals to the results, kriging does not provide the distribution for the errors. At best, some results can be obtained by assuming that the estimate and the kriging variance are the two parameters of a distribution for the error, the rule being the normal distribution. It is not possible to verify such an assumption, which leaves in limbo the significance of the modeling. Third, only under multivariate normality is the kriging variance a good approximation to the error variance. Kriging, however, has not been completely abandoned; it remains at the core of several of the improved methodologies that have been developed to correct these drawbacks.

We use sequential simulation (e.g., Zanon and Leuangthong 2005) in the modeling of all attributes relevant to the assessment of a multi-layer coal deposit because it is a satisfactory formulation to characterize coal beds and to take into account multiple sources of partial information. Using an average coal density leaves net thickness as the primary spatial attribute in tonnage calculations. However, if sufficient density data are available, they can easily be added to the modeling; ignoring density fluctuations does not take any generality away from the present approach. Sequential simulation numerically models surfaces or volumes by assigning values to a regular grid of locations. By design, the results are not unique. For any given attribute partially known through limited drilling, sequential simulation can generate multiple plausible results—maps in our case—employing the same empirical data. Each one of these maps has the same probability to be the true unknown surface. These maps are jointly used to numerically simulate the uncertainty of the attribute of interest at each equally spaced cell in a grid or for total values over a study area. Sequential Gaussian simulation (e.g., Nowak and Verly 2005) was used for the modeling of continuous attributes, such as coal bed thickness or elevation of the base of the oxidation zone. For categorical variables, such as the presence or the absence of a coal bed at any given site, we selected sequential indicator simulation (e.g., Deutsch 2006). Today, user friendly software, such as the public domain SGeMS used here (Remy et al. 2009), allows application of the geostatistical methods mentioned above and several others.

Once cell maps of resource tonnage are generated for every bed, the final step is the aggregation of all beds, which depends on the degree of stochastic correlation among seams estimated from collocated data (e.g., Schuenemeyer and Gautier 2010). Final results are more easily understood when summarized in the form of maps, probability distributions, and tables. Mean, variance, and length of the percentile interval P5P95 in the final results often stabilize using 40–80 realizations (e.g., de Souza et al. 2004).

The following procedure summarizes all necessary modeling steps:

Procedure

  1. A.

    Set a resolution for the modeling by selecting a cell size and define a regular grid containing a total of C cells extending over the area of interest.

     
  2. B.

    Apply sequential Gaussian simulation to generate at least 100 realizations for the elevation of the base of the oxidation zone.

     
  3. C.
    For every bed:
    1. 1.
      Transform the n thickness data at every location \( {\mathbf{z}}_{j} \) into a presence/absence indicator \( i({\mathbf{z}}_{j} ),j = 1,2, \ldots ,n \)
      $$ i({\mathbf{z}}_{j} ) = \left\{ {\begin{array}{ll} {0,} & {\text{if\;the\;bed\;is\;missing}} \\ {1,} & {\text{if\;thickness} > 0} \\ \end{array} } \right.. $$
       
    2. 2.

      Krige the indicator values, prepare a standard deviation map, and use the map to set a maximum value. The outer bounds of modeling will be defined by all cells below the threshold. Cells above the threshold will be blanked in all realizations.

       
    3. 3.

      Use the indicator data to model no coal areas inside the model boundary by applying sequential indicator simulation to generate N realizations, with N being to at least equal 100.

       
    4. 4.

      Apply sequential Gaussian simulation for generating N realizations of thickness.

       
    5. 5.

      Randomly pair each thickness realization with one indicator realization. Then, cell by cell, eliminate from the modeling all cells for which the indicator is 0. The resulting maps are realizations of the locations where the bed is present today after non-deposition and erosion.

       
    6. 6.

      Based on the bed elevation data, generate N sequential Gaussian realizations for the bed top elevation.

       
    7. 7.
      Associate every one of the realizations in the previous Step to one of the oxidation maps in Step B. For every pair of realizations, make a cell-by-cell comparison to generate an oxidation indicator \( o(c),c = 1,2, \ldots ,C \)
      $$ o(c) = \left\{ {\begin{array}{ll} {0,} & {\text{if\;bed\;roof\;elevation} > {\text{elevation\;for\;the\;oxidized\;zone\;base}}} \\ {1,} & {\text{otherwise}} \\ \end{array} } \right.. $$
       
    8. 8.

      Further eliminate from the N realizations in Step C.5 all cells for which o(c) is 0 to discard cells of no commercial value. The resulting maps are the N realizations of thickness after considering non-deposition, erosion, and oxidation.

       
    9. 9.

      Convert cell thickness to cell tonnage using a constant conversion factor.

       
    10. 10.

      If this is not the last bed to model, go back to Step C.3.

       
     
  4. D.

    Using collocated thickness data, calculate the correlation coefficient between every possible pair of beds.

     
  5. E.

    Use the results in the previous Step to aggregate the resources cell by cell.

     
  6. F.

    Use the N realizations resulting from the previous Steps to summarize the results.

     
  7. G.

    End.

     

SGeMS (Remy et al. 2009) was supplemented with utility programs to handle procedures such as determination of the weathered zone and aggregation of resources. The next two sections illustrate the application of the procedure to a site in the Gulf Coast basin of Louisiana. Although the reader will not be able to reproduce our results without adequate geostatistical background or software, everyone should be able to follow and understand the results. This article is not a tutorial in geostatistics; the reader is encouraged to consult the references for a more in-depth understanding of geostatistics (e.g., Olea 2009).

Geology of a Test Site

The Gulf Coast site chosen to illustrate the procedure was selected for its moderately complex geology, presence of three coal beds, and the availability of three sets of drilling data: reconnaissance, initial infill, and development (Figs. 1, 2). We provide only a general description of the geology and cannot release the exact location because the dataset is confidential. The coal beds dip to the east and northeast, so economic interest decreases in those directions beyond the boundaries of the study area due to excessive cover. According to additional drilling not shown here, the coal beds do not continue in any of the other directions. The spacing in the reconnaissance stage was 2–3 and 0.5–0.75 mi for the infill; and approximately 0.25 mi at the development stage. This progressive increase in drilling density offers the possibility of comparing models based on early drilling against additional exploratory drilling, a critical step for validating any methodology.
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Figure 1

Drill hole locations indicating progression in drilling and detail in the elevation for the base of the oxidation zone, with coordinates in thousands of feet: (a) reconnaissance, (b) infill, and (c) development

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Figure 2

Development drilling success for the three beds giving an idea of their extension. The solid black line denotes location of cross section in Figure 3. (a) B5 bed, (b) B bed, and (c) C bed

Figure 3 shows a representative cross section selected to glimpse the coal bed geometry in the vertical direction. The depth of oxidation is highly variable for this deposit. For example, the depth of oxidation for the cross section ranges from 6 to 47 ft (Fig. 3). The extent of the coal beds is controlled by a combination of erosion (topography) and non-deposition (paleochannels) that shaped the original peat deposition. All the three coal beds crop out along the western side of the deposit between drill holes 1 and 2 in Figure 3. Gaps in the continuity of the coal beds indicate the presence of paleochannels. A channel marked by drill holes 7 and 8 impacted all three beds, whereas only the C bed was missing in drill holes 14 and 15, denoting a spatiotemporal varying dynamic in the evolution of the deposit departing from a regular stacking of coal beds.
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Figure 3

East–west cross section along the central part of the study area. Significance of colors annotated at left and right ends

Geological Modeling of the Case Study

Modeling bed by bed and aggregating the partial results at the end yield more geological details and more precise results than aggregating the data up front. With three coal beds and an equal number of drilling stages, modeling of all the stages and layers in the deposit required the preparation of nine sets of partial results. Modeling using only the reconnaissance data is the hardest to do correctly because of the scant amount of information available. Modeling based on the reconnaissance drilling of deepest coal bed B5 has been selected for illustrating the methodology in detail. Data from the infill and development stages were used solely to check the results of the reconnaissance modeling.

In a situation like ours with multiple datasets with different spacings, the densest is the one to analyze for deciding a systematic grid cell spacing (Step A). A square grid cell size of 200 ft was selected based on experience gained modeling over 50 lignite deposits in the Gulf Coast and heuristic rules often cited in the literature: (a) the spacing must be smaller than the average distance among drill holes, (b) at least 95% of the drill holes must land in different cells, and (c) the resolution of the grid should be sufficient, without being excessive, to model all details relevant to the assessment (e.g., Jones et al. 1986; Deutsch 2002; Hengl 2007).

Figure 4 displays the thickness data for coal bed B5 for both the initial and the final drilling stages. After converting the thickness values to indicators (Step C.1), Figure 5 shows the data and its kriging standard deviation map. Upon analyzing this result, the effective study area was limited to those cells where the standard deviation is at most equal to 0.55 (Step C.2). A value of 0.5 is a good default value. Figure 6 displays two possible scenarios of no coal areas for bed B5 according to a modeling of the same indicator data applying sequential indicator simulation (Step C.3). Figure 7 shows two thickness realizations of bed B5 for the extents depicted in Figure 6a, b where the bed was deposited and at least partly preserved (Steps C.4 and C.5).
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Figure 4

Thickness data for coal bed B5. Coordinates are in thousands of feet: (a) posting of reconnaissance drill holes, (b) histogram of reconnaissance thickness data, (b) development drill hole posting, and (d) histogram of development thickness data

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Figure 5

Reconnaissance thickness indicators for coal bed B5. (a) Posting of values, (b) kriging standard deviation truncated at 0.55

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Figure 6

First two realizations for presence/absence of bed B5 out of a total of 100 generated. They honor the data in Figure 5a

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Figure 7

First two of 100 realizations of B5 bed coal thickness

The following combination of steps is intended to discard those cells where the coal has been weathered. Figure 8a shows one of the 100 realizations obtained using the data in Figure 1a (Step B). This oxidation surface is the same for all the three coal beds. When this surface is combined with the realizations for the elevation of the top of B5 such as the one shown in Figure 8b (Step C.6), it is possible to obtain the indicator realizations, the first one being the one shown in Figure 8c (Step C.7). Figure 8d is one of the 100 thickness realizations (Step C.8).
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Figure 8

Modeling of the oxidation effect for the first of 100 realizations. (a) Elevation for the base of the oxidized zone, (b) elevation for the B5 bed roof, (c) oxidation indicator, (d) coal thickness realization in Figure 7a less oxidation zone

Figure 9 is a final rendition of cell tonnage maps after performing Step C.9 using a conversion factor of 1.607 to report the weight in thousands of short tons per cell. This calculation presumes a coal density conversion factor of 1,750 short tons per acre-feet. Figures 10 and 11 are obtained after repeating Step C for coal beds B and C.
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Figure 9

B5 cell tonnage maps based on 100 realizations. (a) Minimum resources (90.3 × 106 s-ton), (b) median resources (135.2 × 106 s-ton), and (c) maximum resources (182.6 × 106 s-ton)

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Figure 10

B cell tonnage maps based on 100 realizations. (a) Minimum resources (243.3 × 106 s-ton), (b) median resources (334.7 × 106 s-ton), (c) maximum resources (448.4 × 106 s-ton)

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Figure 11

C cell tonnage maps based on 100 realizations. (a) Minimum resources (66.9 × 106 s-ton), (b) median resources (113.8 × 106 s-ton), and (c) maximum resources (176 × 106 s-ton)

Even when considering only pairs of thickness values larger than zero, Figure 12 shows that the correlation coefficient between beds is very low (Step D). Thus, in Step E, the correct way to aggregate the resources of the three beds, in this case, is by pairing the realizations at random—i.e., without paying attention to the total tonnage per bed—and then adding collocated cell values. In our modeling, there were 100 groups of three realizations, one per bed. Should the beds by partly correlated, the correlation can be imposed using the Cholesky decomposition (e.g., Schuenemeyer 2005). The aggregated realizations partly rendered in Figure 13 plus the realizations for each individual bed partly displayed in Figures 911 are the basic elements for assessing the coal resources and their associated uncertainty.
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Figure 12

Bed thickness correlation. (a) C–B, 29 pairs with correlation coefficient of 0.14, (b) C–B5, 29 pairs with correlation coefficient of 0.13, and (c) B–B5, 28 pairs with a correlation coefficient of 0.03

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Figure 13

Two forms of display for three representative results out of 100 realizations. (a) Three-bed aggregation that resulted in the minimum aggregated tonnage (484.6 × 106 s-ton) with (b) showing the number of beds, (c) median aggregated tonnage (598.9 × 106 s-ton) with (d) showing the number of beds, (e) maximum aggregated tonnage (725.6 × 106 s-ton) with (f) showing the number of beds

Stochastic Resource Assessment of the Case Study

The realizations for each individual layer and the aggregated tonnage of the three beds contain abundant information that offers a variety of alternatives for reporting the resources obtained by the geologic modeling and associated uncertainty. For example, Figure 14a is a histogram for the 100 realizations partly displayed in Figure 13. By applying the procedure to the infill and development data, it is possible to model the sensitivity of results to the drilling density. As expected from any adequate modeling working with unbiased samples, there is agreement about the central tendency. Discrepancies between the reconnaissance and the other means are approximately 2%. By contrast, differences in dispersion are significant. Any of the numerical distributions can be used to obtain the probability associated with any confidence interval. The easiest ones to obtain are those that can be calculated directly from the listed percentiles. For example, in Figure 14a, there is a 90% that the true tonnage will be between 519.8 and 669.6 million s-ton.
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Figure 14

Frequency distribution of total cell-based aggregated resources based on (a) reconnaissance drilling, (b) infill drilling, and (c) development drilling

Whereas Figure 14 summarizes uncertainty about resources in the entire deposit, Figure 15 shows the probability distribution for a single cell in the study area. The possibilities that geostatistics offers at the cell level are more numerous and equally sound theoretically. At the cell scale, the information comes from the probability distributions that can be numerically prepared at every cell. Instead of adding all cells in a realization, attention can be focused on a cell location. By doing so, it can be seen that each realization contributes one plausible tonnage value at the given cell. The actual number of values can be less than the number of realizations if there is no coal bed at some of the realizations. The set of values numerically models the uncertainty about the true tonnage at a given location. Figure 15 illustrates, for the reconnaissance modeling, the case of the cell whose center is at (30.0, 45.8). At this cell, four realizations predicted no coal for the three beds, implying a 0.96 probability—on a scale of 0–1—that there is indeed some coal in such a cell. Like the global distribution, the cell distribution contains all the uncertainty information that can be extracted from any probability distribution. Cell (30.0, 45.8) was one of the locations drilled during the development stage, penetrating 1.9 ft of beds B and B5 and 2 ft of coal C, for a total of 5.8 ft, which is equivalent to 9.3 thousand s-ton. Considering that the median of the distribution is 10.1 thousand s-ton, the actual value is quite close to the center of the distribution, although that is not always the case. What is intended in stochastic simulation is that the actual value indeed falls inside the interval defined by the extreme values and that the predicted confidence intervals over the entire deposit agree on average with data collected after completion of the modeling (Olea 2012). Figure 16 compares the reconnaissance modeling to the development drilling data and checks that, in fact, over all cells, there is concordance between predicted probabilities and the actual proportion of measurements falling below any threshold. In the example, the largest discrepancy was for 100%, in which case only 95% of the observations were below the maximum value in the cell distributions. On average, the discrepancy for any other probability was at most 2 percentage points, which indicates that the cell distributions indeed provide on average a reliable and unbiased account about expected cell tonnages.
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Figure 15

Numerical model for the uncertainty in tonnage of all the three coal beds for the cell at (30.0, 45.8) near the center of the study area, based on the reconnaissance thickness data. The red arrow points to the tonnage calculated from the actual measurements in the development drill hole for that cell

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Figure 16

Scatterplot showing global degree of agreement between predicted probabilities based on cell distributions using reconnaissance data of coal bed B and actual outcomes according the development drilling. The segmented line denotes perfect agreement between reality and modeling

Any of the statistics in Figure 15 and others not listed there, say, skewness, can be collectively displayed on a map. Figure 17 renders the case of two statistics measuring the dispersion of the values. For both, the ideal value is zero, which denotes no uncertainty. This situation, which happens only at the drilled cells, is not noticeable because of the map resolution and because of sudden increase in uncertainty even for the cell without a drill hole that are adjacent to a drilled cell. Measures of dispersion can be used as a discriminant to classify the cells according to uncertainty.
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Figure 17

Summary statistics mapping dispersion in the cell probability distributions. (a) Standard deviation and (b) length of the interval P5P95

Figure 18 illustrates the case of reconnaissance cell tonnage using the cell standard deviation that, being the standard deviation of an estimate, it is also known as standard error. For brevity, the resources were segregated into three classes, but the assessor is free to set the number of classes and its boundaries in any way that serves an economic or mining purpose. For this presentation, we set the boundaries in such a way that the number of cells going into each bin is exactly the same: 31351. Certainly, the sum of the resources in Figure 18a–c is equal to those in Figure 14a. Despite the equality in number of cells, Figure 18 shows a proportional effect well known in mining: the larger the resources at a cell, the larger its uncertainty (e.g., Manchuk et al. 2009). At the same time, note that the larger the resources associated with a class, the larger the uncertainty about the tonnage per class. In Figure 17, dispersion is larger to the east because geographically the resources are the highest there.
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Figure 18

Classification of cell tonnage according uncertainty. (a) Cell standard error <3.0 × 106 s-ton, (b) cell standard error between 3.0 × 106 and 4.0 × 106 s-ton, and (c) cell standard error >4.0 × 106 s-ton

Figure 19 offers an alternative display of the same information used in the preparation of the previous histograms. The curve instantaneously provides the amount of resources at any interval or below any level of uncertainty and thus offers great flexibility. Relative to Figure 18, the main disadvantage is loss of the form of the distribution of values at an uncertainty class. Still, for a first impression about any deposit, cumulative curves are a good synoptic way to summarize and link the amount of total resources to their local uncertainty.
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Figure 19

Cumulative tonnage and number of cells as a function of the cell standard error. The segmented lines mark the class boundaries in Figure 18

Conclusions

Stochastic simulation allows both a detailed geological characterization of coal deposits and a quantification of associated uncertainties inherent to the modeling. Geological factors considered in the modeling can be bed boundaries, erosion, oxidation, bed thickness, and coal density.

The modeling provides detailed as well as synoptic results for uncertainties at both the deposit and the cell level, as well as either for total resources or for resources per bed. For total resources, a probability distribution provides likely total resources and associated probabilities. At the cell level, it is possible to map cell tonnage, generate probability distributions cell by cell, and summarize the expected cell tonnage in the form of a cumulative distribution as a function of cell uncertainty. These criteria are more informative and theoretically sounder than those put forth in U.S. Geological Survey Circular 891 (Wood et al. 1983).

Acknowledgments

We are grateful to Lawrence Drew (U.S. Geological Survey) and Gerald Weisenfluh (Kentucky Geological Survey) for reviewing a preliminary version of the manuscript and making suggestions for its improvement. Two anonymous reviewers contributed additional suggestions as part of the evaluation of the article by the journal.

Copyright information

© International Association for Mathematical Geology (outside the USA) 2012