# Matheronian geostatistics—Quo vadis?

- 206 Downloads
- 27 Citations

## Abstract

Components of geostatistical estimation, developed as a method for ore deposit assessment, are discussed in detail. The assumption that spatial observations can be treated as a stochastic process is judged to be an inappropriate model for natural data. Problems of semivariogram formulation are reviewed, and this method is considered to be inadequate for estimating the function being sought. Characteristics of bivariate interpolation are summarized, highlighting kriging limitations as an interpolation method. Limitations are similar to those of inverse distance weighted observations interpolation. Attention is drawn to the local bias of kriging and misplaced claims that it is an “optimal” interpolation method. The so-called “estimation variance,” interpreted as providing confidence limits for estimation of mining blocks, is shown to be meaningless as an index of local variation. The claim that geostatistics constitutes a “new science” is examined in detail. Such novelties as exist in the method are shown to transgress accepted principles of scientific inference. Stochastic modeling in general is discussed, and purposes of the approach emphasized. For the purpose of detailed quantitative assessment it can provide only prediction qualified by hypothesis at best. Such an approach should play no part in ore deposit assessment where the need is for local detailed inventories; these can only be achieved properly through local deterministic methods, where prediction is purely deductive.

### Key words

deterministic “estimation variance” interpolation geostatistics kriging least-squares prediction ore deposit assessment probabilistic semivariogram statistical inference## Preview

Unable to display preview. Download preview PDF.

### References

- Akima, H., 1975, Comments on ‘optimal contour mapping using universal kriging’ by Richardo A. Olea: J. Geophys. Res., v. 80, no. 5, p. 832–834.Google Scholar
- Akima, H., 1978, A method of bivariate interpolation and smooth surface fitting for irregularly distributed data points: ACM Trans. Math. Software, no. 4, p. 148–159.Google Scholar
- Armstrong, M. 1983, Comparing drilling patterns for coal reserve assessment: Proceedings of the Australas Institute of Mining and Metallurgy, no. 288, p. 1–5.Google Scholar
- Armstrong, M., 1984a, Problems with universal kriging: Math. Geol., v. 16, no. 1, p. 101–108.Google Scholar
- Armstrong, M., 1984b, Common problems seen in variograms: Math. Geol., v. 16, no. 3, p. 305–313.Google Scholar
- Bagnold, R. A., 1983, The nature and correlation of random distributions: Proc. Roy. Soc. Lond. Ser. A, v. 388, p. 273–291.Google Scholar
- Brooker, P. I., 1983, Semi-variogram estimation using a simulated deposit: Min. Eng., p. 37–42.Google Scholar
- Brown, B. W., 1961, Stochastic variables of geologic search and decision: Bull. Geol. Soc. Amer., v. 72, p. 1675–1686.Google Scholar
- Chalmers, A. F., 1978, What is this thing called science?: University of Queensland Press, Brisbane, 157 p.Google Scholar
- Clark, I., 1979, Practical geostatistics: Applied Science Publishers, London, 129 p.Google Scholar
- Cressie, N. and Hawkins, D. M., 1980, Robust estimation of the variogram. I.: Math. Geol., v. 12, no. 2, p. 115–125.Google Scholar
- David, M., 1977, Geostatistical ore reserve estimation: Elsevier Scientific Publishing Company, Amsterdam, 364 p.Google Scholar
- David, M., 1979, Grade and tonnage problems,
*in*A. Weiss (Ed.) Computer methods for the 80s in the mineral industry: American Institute of Mining, Metallurgical and Petroleum Engineers, New York, p. 170–189.Google Scholar - Davis, M. W. D. and David, M., 1978, Automatic kriging and contouring in the presence of trends (universal kriging made simple): J. Can. Petrol. Tech., v. 17, p. 90–98.Google Scholar
- Davis, M. W. D. and Grivet, C., 1984, Kriging in a global neighbourhood: Math. Geol., v. 16, no. 3, p. 249–265.Google Scholar
- De Wijs, H. J., 1951, Statistics of ore distribution. I.: Geol. Mijnbouw, v. 15, no. 11, p. 365–375.Google Scholar
- De Wijs, H. J., 1953, Statistics of ore distribution. II.: Geol. Mijnbouw, v. 15, no. 1, p. 12–14.Google Scholar
- Dubrule, O., 1984, Comparing splines and kriging: Comput. Geosci. v. 10, p. 327–338.Google Scholar
- Fisher, R. A., 1959, Statistical methods and scientific inference: Oliver Boyd, Edinburgh and London, 175 p.Google Scholar
- Forristall, G. Z., 1978, On the statistical distribution of wave heights in a storm: J. Geophys. Res., v. 83, no. C5, p. 2353–2358.Google Scholar
- Foster, R. J. and Philip, G. M., 1976, Statistical analysis of the Tertiary holasteriod echinoid
*Corystus dysasteroides*from Australasia: Thalassia Jugoslav., v. 12, no. 1, p. 129–184.Google Scholar - Gordon, W. J. and Wixom, J. A., 1978, Shepard's method of “metric interpolation” and bivariate and multivariate interpolation: Math. Comput., v. 32, p. 253–264.Google Scholar
- Harding, J. E., 1923, How to calculate tonnage and grade of an ore body: Eng. Min. J., p. 445–448.Google Scholar
- Hardy, R. L., 1977, Least squares prediction: Photogram. Eng. Remote Sens. v. 18, no. 4, p. 475–492.Google Scholar
- Haring, R. E. and Heideman, J. C., 1978, Gulf of Mexico rare wave return periods: Proceedings of the 10th Offshore Technology Conference, Houston, Texas, p. 1537–1544.Google Scholar
- Huber, P. J., 1973., Robust regression: Asymtotics, conjectures and Monte Carlo: Ann. Stat., v. 1, p. 799–821.Google Scholar
- Huijbregts, C. and Matheron, G., 1970, Universal kriging—an optimal approach to trend surface analysis,
*in*Decision-making in the mineral industry: Can. Inst. Min. Metall., Special volume, no. 12, p. 159–169.Google Scholar - Journel, A. G., 1980, The lognormal approach to predicting local distributions of selective running unit grades: Math. Geol., v. 12, no. 4, pp. 285–303.Google Scholar
- Journel, A. G., 1985, The deterministic side of geostatistics: Math. Geol., v. 17, no. 1, p. 1–15.Google Scholar
- Journel, A. G. and Huijbregts, C. J., 1978, Mining geostastics: Academic Press, London, 600 p.Google Scholar
- Jowett, G. H. 1955, The comparison of means of industrial time series: Appl. Stat., v. 4, p. 32–46.Google Scholar
- King, H. F., McMahon, D. W., and Buijtor, G. J., 1982, A guide to the understanding of ore reserve estimation: Proc. Austral. Inst. Min. Metall., v. 281 (supplement), 21 p.Google Scholar
- Knudsen, H. P., Kim, Y. C., and Mueller, E., 1978, Comparative study of the geostatistical ore reserve estimation method over the conventional methods: Min. Eng., p. 54–58.Google Scholar
- Krige, D. G., 1962, Statistical applications in mine valuation: J. Inst. Mine Surv, South Afr., v. 12, no. 2, p. 45–84 and v. 12, no. 3, p. 95–136.Google Scholar
- Krige, D. G., 1966, Two-dimensional weighted moving trend surfaces for ore valuation: Symposium on Mathematical Statistics and Computer Applications in Ore Valuation: South Afr. Inst. Min. Metall., p. 13–38.Google Scholar
- Krige, D. G., 1981, Lognormal—de Wijsian geostatics for ore evaluation Monograph, Geostatistics 1 (2nd ed.): South Afr. Inst. Min. Metall., 51 p.Google Scholar
- Kuhn, T. 1962, The structure of scientific revolutions: Chicago University Press, Chicago, Illinois.Google Scholar
- Lancaster, P. and Salkauskas, K., 1981, Surfaces generated by moving least squares methods: Math. Comput., v. 37, p. 141–158.Google Scholar
- Longuet-Higgins, M. S., 1952, On the statistical distribution of the heights of sea waves: J. Mar. Res., v. 11, no. 3, p. 245–266.Google Scholar
- McLain, D. H., 1974, Drawing contours from arbitrary data points: Comput. J., v. 17, p. 318–324.Google Scholar
- Matheron, G., 1962, Traité de geostatistique appliquée, v. 1 (1962), 334 p., vol. 2 (1963), 172 p. Editions Technip, Paris.Google Scholar
- Matheron, G., 1963, Principles of geostatistics: Econ. Geol., v. 58, p. 1246–1266.Google Scholar
- Matheron, G., 1973, The intrinsic random functions and their applications: Adv. Appl. Prob., v. 5, p. 439–468.Google Scholar
- Myers, D. E., Begovich, C. L., Butz, T. R., and Kane, V. E., 1982, Variogram models for regional groundwater geochemical data: Math. Geol. v. 14, no. 6, p. 629–644.Google Scholar
- Narula, S. C. and Wellington, J. F., 1982, The minimum sume of absolute errors regression: A state of the art survey: Inter. Stat. Rev., v. 50, p. 317–326.Google Scholar
- Olea, R. A., 1974, Optimal contour mapping using universal kriging: J. Geophys. Res., v. 79, no. 5, p. 695–702.Google Scholar
- Olea, R. A., 1984, Systematic sampling of spatial functions: Kansas Series on Spatial Analysis., no. 7: Kansas Geological Survey, University of Kansas, Lawrence, Kansas, 57 p.Google Scholar
- Pauncz, I. and Nixon, T. R., 1980, Application of geostatistics for a more precise statement of coal researches: Aust. Coal Indust. Res. Lab. Report, no. 3, 76 p.Google Scholar
- Philip, G. M. and Watson, D. F. 1982a, A precise method for determining contoured surfaces: Aust. Petrol. Expl. Assoc. J., v. 22, no. 1, p. 205–212.Google Scholar
- Philip, G. M. and Watson, D. F., 1982b, Optimum drilling patterns for establishing coal reserves: Aust. J. Coal Min. Tech. Res., v. 2, p. 65–68.Google Scholar
- Philip, G. M. and Watson, D. F., 1984, Drilling patterns and ore reserve assessments: Proc. Australas, Inst. Min. Metall., no. 289, p. 205–211.Google Scholar
- Philip, G. M. and Watson, D. F., 1985a, A method for assessing local variation among scattered measurements: Math. Geol. (in press).Google Scholar
- Philip, G. M. and Watson, D. F., 1985b, Theoretical aspects of grade-tonnage calculations: Inter. J. Min. Eng., v. 3 (in press).Google Scholar
- Philip, G. M. and Watson, D. F., 1985c, A deterministic approach to computing ore reserves. I. Grade estimation: Occasional Publ. 2, Earth Resources Foundation, University of Sydney.Google Scholar
- Philip, G. M. and Watson, D. F., 1985d, Some limitations in the geostatistical evaluation of ore deposits: Inter. J. Min. Eng., v. 3 (in press).Google Scholar
- Philip, G. M. and Watson, D. F., 1985e, Comment on ‘Comparing splines and kriging’ by Oliver Dubrule: Comput. Geosci. (in press).Google Scholar
- Popper, K. and Miller, D., 1983, A proof of the impossibility of inductive probability: Nature, v. 302, p. 687–688, and v. 310, p. 434.Google Scholar
- Poppoff, C. C., 1966, Computing reserves for mineral deposits: Principles and conventional methods: U.S. Bureau of Mines, Information Circular, 8283, 113 p.Google Scholar
- Quine, M. P. and Watson, D. F., 1984, Radial generation of
*n*-dimensional Poisson point processes: J. Appl. Prob., v. 21, p. 548–557.Google Scholar - Rendu, J.-M., 1980, Disjunctive kriging; Comparison of theory with actual results: Math. Geol., v. 12, no. 4, p. 305–320.Google Scholar
- Royle, A. G., Clausen, F. L., and Frederiksen, P., 1981, Practical universal kriging and automatic contouring: Geoprocessing, v. 1, p. 377–394.Google Scholar
- Rutledge, R. W., 1975, The potential of geostatistics in the development of mining:
*in*Guarascio, M., David, M., and Huijbregts, C. (Eds), Advanced geostatistics in the mining industry: D. Reidel, Dordrecht, Holland, p. 295–311.Google Scholar - Sager, T. W., 1983, Estimating modes and isopleths: Commun. Stat. Theor. Meth., v. 12, no. 5.Google Scholar
- Sahu, B. K., 1982, Stochastic modelling of mineral deposits: Min. Depos., v. 17, p. 99–105.Google Scholar
- Schagen, I. P., 1979, Interpolation in two dimensions—A new technique: J. Inst. Math. Appl., v. 23, p. 53–59.Google Scholar
- Schwarzacher, W., 1975, Sedimentation models and quantitative stratigraphy: Elsevier Scientific Publishing Company, Amsterdam, 382 p.Google Scholar
- Shurtz, R. F., 1984, A stochastic aberration—The theory of regionalised variables: unpublished manuscript, 13 p.Google Scholar
- Sibson, R., 1980, A brief description of natural neighbor interpolation;
*in*B. Barnett (Ed.) Interpreting multivariate data: John Wiley & Sons, New York, p. 21–26.Google Scholar - Singer, D. A. and De Young, J. H., 1980, What can grade tonnage relations really tell us?
*in*C. Guillemin and P. Lagney (Eds.), Resources minerales—Mineral resources Bur. Recher. Geol. Min. Mem., no. 106, p. 91–101.Google Scholar - Skinner, B. J., 1976, A second iron age ahead?: Amer. Sci., v. 64, no. 3, p. 258–269.Google Scholar
- Sluijk, D. and Parker, J. R., 1985, Comparison of pre-drilling prediction and post drilling outcome, using Shell's PACQ method,
*in*Rice and James (Eds.), Oil and gas assessment—Methods and applications: Amer. Assoc. Pet. Geol. Mem., no. 40 (in press).Google Scholar - Smith, F. G. and Watson, D. F., 1972, A minimal test of significance of nonuniform density of data points in a three variable closed array: Can. J. Earth Sci. v. 9, no. 9, p. 1124–1128.Google Scholar
- Surken, A. J., Denny, J. R., and Batcha, J., 1964, Computer contouring: A new tool for evaluating and analysis of mines: Eng. Min. Jour., no. 165, p. 72–76.Google Scholar
- Vistelius, A. B., 1960, The skew frequency distribution and the fundamental law of geochemistry: J. Geol. v. 68, p. 1–22.Google Scholar
- Wang, C. Y., 1983,
*C*^{1}rational interpolation over an arbitrary triangle: Comput.-aid. Des., v. 15, no. 1, p. 33–36.Google Scholar - Watson, D. F., 1981, Computing the
*n*-dimensional Delaunay tessellation with application to the Voronoi polytopes: Comput. J., v. 24, p. 167–172.Google Scholar - Watson, D. F., 1983, Two images for three dimensions: Prac. Comput., Aug., p. 104–107; Errata, Sept., p. 8.Google Scholar
- Watson, D. F., 1985, Natural neighbour sorting: Aust. Comput. J., v. 17, no. 4 (in press).Google Scholar
- Watson, D. F. and Philip, G. M. 1984a, Systematic triangulations: Comput. Vis. Graph. Imag. Process., v. 26, p. 217–223.Google Scholar
- Watson, D. F. and Philip, G. M., 1984b, Triangle based interpolation: Math. Geol., v. 16, p. 779–795.Google Scholar
- Watson, D. F. and Philip, G. M., 1985, A refinement of inverse distance weighted interpolation: Geoprocessing (in press).Google Scholar
- Watson, G. S., 1971, Trend surface analysis: Math. Geol., v. 3, p. 215–226.Google Scholar
- Whitten, E. H. T., 1966, The general linear equation in prediction of gold content in Witswaters-rand rocks, South Africa: Symposium on Mathematical Statistics and Computer Applications: South Afr. Inst. Min. Metall., p. 125–141.Google Scholar
- Whitten, E. H. T., 1977, Stochastic models in geology: J. Geol., v. 85, p. 321–330.Google Scholar
- Whittle, P., 1954, On stationary processes in the plane: Biometrika, v. 41, p. 434–449.Google Scholar
- Ziolkowski, A., 1982, Further thoughts on Popperian geophysics: Geophys. Prospect., v. 30, p. 155–156.Google Scholar