Considering the Epistemic Uncertainties of the Variogram Model in Locating Additional Exploratory Drillholes

Document Type: Research Paper

Authors

Department of Mining Engineering, University of Kashan, Kashan, Iran

Abstract

To enhance the certainty of the grade block model, it is necessary to increase the number of exploratory drillholes and collect more data from the deposit. The inputs of the process of locating these additional drillholes include the variogram model parameters, locations of the samples taken from the initial drillholes, and the geological block model. The uncertainties of these inputs will lead to uncertainties in the optimal locations of additional drillholes. Meanwhile, the locations of the initial data are crisp, but the variogram model parameters and the geological model have uncertainties due to the limitation of the number of initial data. In this paper, effort has been made to consider the effects of variogram uncertainties on the optimal location of additional drillholes using the fuzzy kriging and solve the locating problem with the genetic algorithm (GA) optimization method.A bauxite deposit case study has shown the efficiency of the proposed model.

Keywords


1] Kim, Y.C., Martino, F. & Chopra, I.K. (1981). Application of geostatistics in a coal deposit. Journal Name: Min. Eng. (Littleton, Colo.); (United States); Journal Volume: 33:10, Medium: X; Size: p. 1476-1481.
[2] Walton, D.R. & Kauffman, P. W. (1982). Some Practical Considerations in Applying Geostatistics to Coal Reserve Estimation. Paper presented at the SME-AIME, Dallas.
[3] Russo, D. (1984). Design of an Optimal Sampling Network for Estimating the Variogram1. Soil Sci. Soc. Am. J., 48(4), 708-716. doi: 10.2136/sssaj1984.03615995004800040003x.
[4] Yfantis, E., Flatman, G. & Behar, J. (1987). Efficiency of kriging estimation for square, triangular, and hexagonal grids. Mathematical Geology, 19(3), 183-205. doi: 10.1007/BF00897746.
[5] Corsten, L.C.A. & Stein, A. (1994). Nested sampling for estimating spatial semivariograms compared to other designs. Applied Stochastic Models and Data Analysis, 10(2), 103-122. doi: 10.1002/asm.3150100205.
[6] Bogaert, P. & Russo, D. (1999). Optimal spatial sampling design for the estimation of the variogram based on a least squares approach. Water Resources Research, 35(4), 1275-1289. doi: 10.1029/1998WR900078.
[7] van Groenigen, J.W. & Stein, A. (1998). Constrained Optimization of Spatial Sampling using Continuous Simulated Annealing. J Environ Qual, 27(5), 1078-1086.
[8] Warrick, A.W. & Myers, D.E. (1987). Optimization of sampling locations for variogram calculations. Water Resources Research, 23(3), 496-500. doi: 10.1029/WR023i003p00496.
[9] Soltani, S. & Hezarkhani, A. (2011). Determination of Realistic and Statistical Value of the Information Gathered from Exploratory Drilling. Natural Resources Research, 20(4), 207-216. doi: 10.1007/s11053-011-9148-y.
[10] Soltani-Mohammadi, S. & Hezarkhani, A. (2013). A Simulated Annealing-Based Algorithm to Locate Additional Drillholes for Maximizing the Realistic Value of Information. Natural Resources Research, 22(3), 229-237. doi: 10.1007/s11053-013-9212-x.
[11] Soltani, S. & Hezarkhani, A. (2009). Additional exploratory boreholes optimization based on three-dimensional model of ore deposit. Archieves of Mining Science, 54(3), 495--506.
[12] Hossein Morshedy, A. & Memarian, H. (2015). A novel algorithm for designing the layout of additional boreholes. Ore Geology Reviews, 67(0): p. 34-42.
[13] Szidarovszky, F. (1983). Multiobjective observation network design for regionalized variables. International Journal of Mining Engineering, 1(4), 331-342. doi: 10.1007/bf00881549.
[14] Morshedy, A.H., Torabi, S. & Memarian, H. (2015). A new method for 3D designing of complementary exploration drilling layout based on ore value and objective functions.
Soltani-Mohammadi & Soltani / Int. J. Min. & Geo-Eng., Vol.49, No.1, June 2015
74
Arabian Journal of Geosciences, 1-21. doi: 10.1007/s12517-014-1754-7.
[15] Soltani, S., Hezarkhani, A., Erhan Tercan, A. & Karimi, B. (2011). Use of genetic algorithm in optimally locating additional drill holes. Journal of Mining Science, 47(1), 62-72. doi: 10.1134/s1062739147010084.
[16] Morshedy, A.H. & Memarian, H. (2015). A novel algorithm for designing the layout of additional boreholes. Ore Geology Reviews, 67(0), 34-42. doi: http://dx.doi.org/10.1016/j.oregeorev.2014.11.012.
[17] Bardossy, A., Bogardi, I. & Kelly, W.E. (1990a). Kriging with imprecise (fuzzy) variograms. I: Theory. Mathematical Geology, 22(1), 63-79. doi: 10.1007/bf00890297.
[18] Bardossy, A., Bogardi, I. & Kelly, W.E. (1990b). Kriging with imprecise (fuzzy) variograms. II: Application. Mathematical Geology, 22(1), 81-94. doi: 10.1007/bf00890298.
[19] Bardossy, A., Bogardi, I. & Kelly, W.E. (1988). Imprecise (fuzzy) information in geostatistics. Mathematical Geology, 20(4), 287-311. doi: 10.1007/bf00892981.
[20] Handcock, M.S. & Stein, M.L. (1993). A Bayesian Analysis of Kriging. Technometrics, 35(4), 403-410. doi: 10.1080/00401706.1993.10485354.
[21] Journel, A.G.H.C.J. (1978). Mining geostatistics. London; New York: Academic Press.
[22] Webster, R. & Oliver, M.A. (2007). Geostatistics for Environmental Scientists: Wiley.
[23] Soltani, S., Hezarkhani, A. & Erhan Tercan, A. (2011). Optimally locating additional drill holes in three dimensions using grade and simulated annealing. Journal of the Geological Society of India, 80(5), 700-706. doi: 10.1007/s12594-012-0195-8.
[24] Burger, H. & Birkenhake, F. (1994). Geostatistics and the Polygonal Method: A Re-Examination. Proceedings”, International Association for Mathematical Geology (IAMG 95) Annual Conference, p. 50-55.
[25] Loquin, K. & Dubois, D. (2010). Kriging and Epistemic Uncertainty: A Critical Discussion. In R. Jeansoulin, O. Papini, H. Prade & S. Schockaert (Eds.), Methods for Handling Imperfect Spatial Information (Vol. 256, pp. 269-305): Springer Berlin Heidelberg.
[26] Loquin, K. & Dubois, D. (2012). A fuzzy interval analysis approach to kriging with ill-known variogram and data. Soft Computing, 16(5), 769-784. doi: 10.1007/s00500-011-0768-2.
[27] Gen, M. & Cheng, R. (1996). Genetic Algorithms and Engineering Design: John Wiley & Sons, Inc.
[28] Taboada, J., Rivas, T., Saavedra, A., Ordóñez, C., Bastante, F. & Giráldez, E. (2008). Evaluation of the reserve of a granite deposit by fuzzy kriging. Engineering Geology, 99: p. 23-30