AGH University of Science and Technology Faculty of Geology, Geophysics and Environmental Protection DEPARTMENT OF ECONOMIC AND MINING GEOLOGY Av. Mickiewicza 30, 30-059 Cracow, Poland CATEGORIZATION OF MINERAL RESOURCES BASED UPON GEOSTATISTICAL ESTIMATION OF THE CONTINUITY OF CHANGES OF RESOURCE PARAMETERS The research project was financed by AGH University of Science and Technology, grant No. 11.11.140.320 and 11.11.140.627 C 0 C +C 0 g(h) h (C +C) 0 2 3 (C +C) 0 1 3 0.05(C +C) 0 R max Inf R max Ind R max M R min Inf R min Ind R min M R max Inf R max Ind R max M R min Inf R min Ind R min M a max a min a max a min g(h) C C 0 Spherical model: g(h) = C + C sph(h/a) 0 a h i N(h) i R(h) i N(h ) max h h max Figure 1. Sample semivariogram of deposit parameter approximated with the spherical model. Explanations: N(h ) – non-random component of i variability for distance h , R(h ) – random component of i i variability for distance h , a – range of semivariogram i (autocorrelation), h – maximum distance for which max autocorrelation is statistically significant, C – nugget 0 variance, C – spatial variance, – maximum share of non-random component of variability, – share of non-random component of variability for distance h i 4. SUMMARY AND CONCLUSIONS The presented geostatistical methodology of determination of the JORC Code's mineral resources categories must be regarded as a preliminary proposal and a material for further discussion. The methodology is currently verified in some Polish mineral deposits. Particularly important seems to be the analysis of other threshold (minimum) values of the shares of non-random variability component, for example: 75% for “measured” and 50% for “indicated” categories as well as the influence of such thresholds on the amounts of resources calculated for particular categories. The practice of geostatistical modeling should involve also the testing of statistical significance of autocorrelation of deposit parameters. Maximum distance between the observation/sampling sites for which the autocorrelation is statistically significant can be accepted as a maximum extent of inferred resource category. Within various categories, the resources can be estimated with kriging procedures. Some problems may be caused by adequately precise contouring of areas occupied by various resources categories if their patterns are complicated. The proposed methodology of mineral resources categorization, which takes advantage of geostatistically described spatial continuity of the values of deposit parameters, should be strengthened by determination of relative (standard) kriging errors of resource estimations for particular categories. The permissible values of these errors accepted as additional criteria can be applied as corrections of already determined categories. REFERENCES Arik, A. (2002). Comparison of Resource Classification Methodologies With a New Approach. 30th APCOM Symposium Proceedings, Phoenix, Arizona. Australian Guidelines for Estimating and Reporting of Inventory Coal, Coal Resources and Coal Reserves. The Coalfields Geology Council of New South Wales and the Queensland Mining Council (edition 2003). Gringarten, E. and C.V. Deutsch (2001). Variogram Interpretation and Modeling. Math. Geology, Vol. 33. No.4, 507-534. Journel, A. G. and Ch. J. Huijbregts (1978). Mining Geostatistics, London Academic Press, 600 pp. Mucha, J. and M. Wasilewska-Błaszczyk (2012). Variability anisotropy of mineral deposits parameters and its impact on resources estimation – a geostatistical approach. Gospodarka Surowcami Mineralnymi, T. 28, z. 4, 113-135. Mucha, J. and M. Wasilewska-Błaszczyk (2014). Categorization of bituminous coal resources based upon the guidelines of the JORC Code and the geostatistics (in Polish: Kategoryzacja zasobów złóż węgla kamiennego w świetle wytycznych do JORC Code i geostatystyki). Górnictwo Odkrywkowe, Wrocław, nr 2-3, 67-73. Mwasinga, P. P. (2001). Approaching resource classification: General practices and the integration of geostatistics. Computer Applications in the Minerals Industries, Xie, Wang&Jiang (eds), Swets&Zeintlinger, Lisse, 97-104. Nieć, M. (2008). International Classifications of Mineral Resources And Reserves. Problems of harmonization (in Polish: Międzynarodowe klasyfikacje zasobów złóż kopalin. Problemy unifikacji). Gospodarka Surowcami Mineralnymi, 24, 2/4, 267-274. Silva, D.S.F. and J.B. Boisvert (2014). Mineral resource classification: a comparison of new and existing techniques. The Journal of the Southern African Institute of Mining and Metallurgy. Vol. 114, 265-273. Sinclaire, A. J. and G. H. Blackwell (2002). Applied Mineral Inventory Estimation. Cambridge University Press, 381. Sobczyk, E. J. and P.W. Saługa (2013). Coal resource base in Poland from the perspective of using the JORC CODE. Conference: Proceedings of the 23rd World Mining Congress, Montreal, Canada. de Souza, L.E., J.F.C.L. Costa and J.C. Koppe (2009). A Geostatistical Contribution to Assess the Risk Embedded in Resource Classification Methods. Iron Ore Conference Perth, WA, 27 - 29 July 2009. The JORC Code (2012 edition) – Australasian Code for Reporting of Exploration Results, Mineral Resources and Ore Reserves prepared by The Australasian Institute of Mining and Metallurgy (AusIMM), Australian Institute of Geoscientists and Minerals Council of Australia. Figure 3B. Exemplary determination of the ranges of resources categories around observation/sampling sites under the conditions of spherical anisotropic model. Explanations: categories: measured - red, indicated - green, inferred - yellow, C - nugget variance, a – ranges of semivariograms for directions of 0 maximum (a ) and minimum variability (a ), and R , R , R - ranges for min max M Ind Inf resources categories: M – measured, Ind – indicated, Inf – inferred Ć ĈĆ ČĆ ĊĆ ÇĆ DĆ ĎĆ ĐĆ ÐĆ EĆ ĈĆĆ Ć ČĆ ÇĆ ĎĆ ÐĆ ĈĆĆ ĈČĆ Į Í ĂOĚĆÅĚĈĆĆÃ Į Í ĂOĚĆÅĚĐDÃ Į Í ĂOĚĆÅĚDĆÃ Į Í ĂOĚĆÅĚČDÃ h/a [%] U (h/a) [%] N U (h=0)=100% N U (h=0)=75% N U (h=0)=50% N U (h=0)=25% N U (h=0)=100% N U (h=0)=75% N U (h=0)=50% N U (h=0)=25% N g(h) h U (h=0)=100% N U (h=0)=75% N U (h=0)=50% N U (h=0)=25% N U (h=0)=0% N MEASURED (66.6%) INDICATED (33.3%) INFERRED (5.0%) Ć ĈĆ ČĆ ĊĆ ÇĆ DĆ ĎĆ ĐĆ ÐĆ EĆ ĈĆĆ Ć ČĆ ÇĆ ĎĆ ÐĆ ĈĆĆ ĈČĆ Į Í ĂOĚĆÅĚĈĆĆÃ Į Í ĂOĚĆÅĚĐDÃ Į Í ĂOĚĆÅĚDĆÃ Į Í ĂOĚĆÅĚČDÃ h/a [%] 0 U (h/a ) [%] N 0 U (h=0)=100% N U (h=0)=75% N U (h=0)=50% N U (h=0)=25% N U (h=0)=100% N U (h=0)=75% N U (h=0)=50% N U (h=0)=25% N MEASURED (66.6%) INDICATED (33.3%) INFERRED (5.0%) g(h) h U (h=0)=0% N Ć ĈĆ ČĆ ĊĆ ÇĆ DĆ ĎĆ ĐĆ ÐĆ EĆ ĈĆĆ Ć ČĆ ÇĆ ĎĆ ÐĆ ĈĆĆ ĈČĆ Į Í ĂOĚĆÅĚĈĆĆÃ Į Í ĂOĚĆÅĚĐDÃ Į Í ĂOĚĆÅĚDĆÃ Į Í ĂOĚĆÅĚČDÃ h/a [%] 0 U (h/a ) [%] N 0 U (h=0)=100% N U (h=0)=75% N U (h=0)=50% N U (h=0)=25% N U (h=0)=100% N U (h=0)=75% N U (h=0)=50% N U (h=0)=25% N MEASURED (66.6%) INDICATED (33.3%) INFERRED (5.0%) g(h) h U (h=0)=0% N Ć ĈĆ ČĆ ĊĆ ÇĆ DĆ ĎĆ ĐĆ ÐĆ EĆ ĈĆĆ Ć ČĆ ÇĆ ĎĆ ÐĆ ĈĆĆ ĈČĆ Į Í ĂOĚĆÅĚĈĆĆ Ã Į Í ĂOĚĆÅĚĐDÃ Į Í ĂOĚĆÅĚDĆÃ Į Í ĂOĚĆÅĚČDÃ h/a [%] 0 U (h/a ) [%] N 0 U (h=0)=100% N U (h=0)=75% N U (h=0)=50% N U (h=0)=25% N U (h=0)=100% N U (h=0)=75% N U (h=0)=50% N U (h=0)=25% N MEASURED (66.6%) INDICATED (33.3%) INFERRED (5.0%) g(h) h U (h=0)=100% N U (h=0)=75% N U (h=0)=50% N U (h=0)=25% N U (h=0)=0% N SPHERICAL EXPONENTIAL GAUSSIAN LINEAR a a 0 a 0 a 0 2 s 2 s 2 s 2 s C 01 C 02 a 1 a 2 Semivariogram g(h) Sill 1 2 d 1 Distance (h) C C +C 0 100% The scheme of determination of particular resources categories for both the isotropic and anisotropic variability models are shown in Figs. 3A and 3B. If autocorrelation is strong the zones corresponding to various categories may overlap and the areas occupied by particular categories may show complicated patterns. The resources of specific categories can be estimated with the point or the polygon kriging when the areas covered by relevant resources categories are known. Efficient tool for such calculations can be e.g., the ISATIS software. The proposed categorization method of mineral resources should be supplemented by additional criteria, e.g., by the accuracy of resource estimations within the areas covered by particular categories. This accuracy can be expressed by a relative, standard kriging error determined with the polygon kriging method. Jacek MUCHA mucha@geol.agh.edu.pl Monika WASILEWSKA-BŁASZCZYK monika.wasilewska@agh.edu.pl Justyna AUGUŚCIK jauguscik@geol.agh.edu.pl Poster presented at the 17th annual conference of the International Association for Mathematical Geosciences, September 5-13, 2015, Freiberg, Germany ( session: 96. Geostatistics for Mineral Resources; presentation code G1103) INTRODUCTION The proper categorization of resources of mineral deposits is one of the most difficult problems of economic geology despite a number of proposed methodologies. An additional obstacle is a number of various systems of resources categorization applied in various countries and regions, e.g.: JORC Code (Australia), SAMREC Code (South Africa), PERC Code (Europe) CIM Standards (Canada), SME Guidelines (United States), UNFC (Nieć, 2008; Sobczyk and Saługa, 2013). From the point of view of investors and stock exchanges, the resource category should reflect credibility and confidence in recognition of continuity of a deposit and estimations of deposit parameters. The existing systems of reporting on resources of mineral raw-materials do not embrace precise directories suitable for categorization of resources. Some suggestions can be found in guides to categorization systems. For example, some indications concerning the hard-coal deposits are contained in the Australian Guidelines for Estimating and Reporting of Inventory Coal, Coal Resources and Coal Reserves - Edition 2003 (Mucha and Wasilewska-Błaszczyk, 2014). As regards the methodology of resources categorization, the expectations of geologists preparing the assessment reports focus mostly on its correctness, simplicity and easy application to geological practice. Referring to the suggestion contained in the JORC Code, the categorization criterion discussed in the following paper is a spatial continuity (autocorrelation) of a representative resources parameter values determined from the semivariogram model. 1. CURRENT METHODOLOGIES OF RESOURCES CATEGORIZATION The comprehensive reviews of current methods of resources categorization together with some new solutions were recently published by Mwasinga (2001), Sinclaire and Blackwell (2002) Arik (2002), Souza et al. (2009) and Silva and Boisvert (2014). Generally, these authors describe two groups of methods: one based upon mutual relationships between the geometries of observation/sampling grids and the area of categorization, and second – the geostatistical approach based upon Matheron's methodology. The Australian JORC Code (2012), quite popular in the world, distinguishes three categories of resources characterized by the confidence in estimations: inferred (low level of confidence), indicated (reasonable level of confidence) and measured (high level of confidence). An important criterion is the recognition level of continuity of geological and quality parameters of a raw material. 2. GEOSTATISTICAL CHARACTERIZATION OF CONTINUITY OF DEPOSIT PARAMETERS The continuity of given deposit parameter is estimated from the values of theoretical semivariograms (models of semivariograms). Low values of semivariograms for distances close to zero express low dissimilarity (high similarity) of given deposit parameter values and, simultaneously, they indicate high continuity of changes of this parameter within deposit body. In order to characterize quantitatively the degree of continuity of given deposit parameter, we must define (Fig. 1):  The nugget variance C , which expresses local 0 variability and, simultaneously, represents the minimum value of random component of variability.  The spatial variance C, which expresses the maximum value of non-random component of variability in the case of models with asymptote.  The range of semivariogram a, which shows the maximum range of autocorrelation of given parameter. 3. PRINCIPLES OF PROPOSED CATEGORIZATION SYSTEM OF MINERAL RESOURCES The simplest solution is the usage of autocorrelation range (semivariogram) of a representative deposit parameter – e.g., abundance of raw material or its usable component – determined with the sample semivariogam model. Despite its appealing simplicity, this methodology has a serious drawback – it does not consider either the local variability of selected deposit parameter (nugget variance) or the strength of autocorrelation of parameter value. If the share of nugget variance (C ) in overall 0 variability (C +C) of given deposit parameter is high it 0 may happen that the autocorrelation of deposit parameter values (expressed by the share of non- random component) will be statistically insignificant even at distances approximating zero (Fig. 2). Hence, the determination of radii of resources categories around observation/sampling sites based upon the formally determined semivariogram ranges is out of sense. In our proposal, the criterion of resources categorization is the relative share of non-random variability component in an overall variability of a representative deposit parameter (U(h ) – i Fig. 1). Taking into account the experience in assessment of Polish mineral deposits and the JORC Code terminology, we arbitrary ascribed to relevant categories the resources which fall within such distances ( R ) around i observation/sampling sites for which the shares of non-random variability component take the following threshold values (Fig. 3A, 3B): — measured resources category (R ) – M U(h ) > 2/3 (>66.6%), i — indicated resources category (R ) – Ind 2/3 > U(h ) > 1/3 (66.6% > U(h ) > 33.3%), i i — inferred resources category (R ) – inf 1/3 > U(h ) > 1/20 (33.3% > U(h ) > 5%). i i Figure 2. Examples of semivariogram models of strongly diversified ranges (a , a ) and nugget 1 2 variances (C , C ); grey area represents the lack of 01 02 statistically significant autocorrelation of deposit parameter. Explanations: 1 – model with short range of semivariogram and high share of non-random component of variability, 2 – model with long range of semivariogram and low share of non-random component of variability (practically, the autocorrelation of the values of deposit parameter is statistically insignificant), d – range of statistical significant 1 autocorrelation of model 1 The ranges of relevant resources categories can be determined quickly and easily (although only approximately) with graphic method based on the plots drawn for the four most commonly applied geostatistical models (Fig. 4). The maximum ranges of resources categories (expressed in units of the ranges of theoretical semivariograms) can be established using the maximum shares of non-random component of parameter variability and its proposed threshold values for particular categories of mineral resources. Figure 4. Diagrams for determination of maximum ranges of resources categories (measured, indicated, inferred) for 4 geostatistical models (spherical, exponential, Gaussian, linear) based on proposed threshold values of share of non- random component of variability Explanations: U (h=0)=U – maximum share of non-random component of variability ( ), a (a ) – range (practical) of N max 0 theoretical semivariogram, h/a (h/a ) – distance expressed in units of range (maximum range of category) 0 U = max C C +C 0 100% U(h )= i N(h ) i C +C 0 100% C +C 0 g(h) (C +C) 0 2 3 (C +C) 0 1 3 0.05(C +C) 0 C 0 R M R Ind R Inf a h R Inf R Ind R M Figure 3A. Exemplary determination of the ranges of particular resources categories around observation/sampling sites (black points) under the conditions of spherical isotropic model. Explanations: categories: measured – red, indicated – green, inferred – yellow, C – 0 nugget variance, C – spatial variance, R , R , R – ranges of categories: M – measured, Ind – indicated, M Ind Inf Inf – inferred, a – range of semivariogram (autocorrelation)