Abstract
By using a method originally developed by P. Whittle, it is shown that a continuous random variable in three-dimensional space has an exponential autocorrelation function if it is subject to a property analogous to the Markov property in time-series analysis.
The problem of estimating autocorrelation functions from irregularly distributed map data is discussed. Approximate autocorrelation functions are shown for a set of 200 subsurface elevations on top of the Arbuckle Group (Cambrian-Ordovician) in Kansas. Trend-surface analysis, a method of kriging and a combination of the two procedures also are applied to the data. Areal interpolation can be done by a method that consists of three steps: (1) fitting a low-order polynomial trend surface; (2) estimation of the autocorrelation function for the residuals; (3) application of kriging to the residuals.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Abramowitz, M., and Stegun, A., 1965, Handbook of mathematical functions: National Bureau of Standards, Applied Math. Ser. 55, 1046 p.
Agterberg, F. P., 1965, The technique of serial correlation applied to continuous series of element concentration values in homogenous rocks: Jour. Geology, v. 73, no. 1, p. 142–154.
Agterberg, F. P., 1966, Trend surfaces with autocorrelated residuals: Pennsylvania State Univ., Mineral Industries, v. 1, p. Hl-H19.
Agterberg, F. P., 1967, Mathematical models in ore eval-uation: Jour. Canadian Operational Res. Soc., v. 5, no. 3, p. 144–158.
Agterberg, F. P., 1968, Application of trend analysis in the evaluation of the Whalesback Mine, Newfoundland: Canadian Inst. Mining Metallurgy, Sp. Vol. 9, p. 77–88.
Agterberg, F. P., 1969, Interpolation of areally distributed data: Colorado Sch. Mines Quart., v. 64, no. 3, p. 217–237.
Bartlett, M. S., 1964, A note on spatial pattern: Biometrics, v. 20, p. 891–892.
Bartlett, M. S., 1966, An introduction to stochastic processes ( 2nd ed. ): Cambridge Univ. Press, 362 p.
Blackman, R. B., and Tukey, J. W., 1959, The measurement of power spectra: Dover Publ., New York, 190 p.
Cox, D. R., and Miller, H. D., 1965, The theory of stochastic processes: John Wiley Sons, New York, 398 p.
Davis, J. C., and Preston, F. W., in press, Optical processing an attempt to avoid the computer.
Esler, J. E., and Preston, F. W., 1967, FORTRAN IV program for the GE 625 to compute the power spectrum of geological surfaces: Kansas Geol. Survey Computer Contr. 16, 23 p.
Good, D. I., 1964, FORTRAN II trend-surface program for the IBM 1620: Kansas Geol. Survey Sp. Dist. Publ. 14, 54 p.
Harbaugh, J. W., and Merriam, D. F., 1968, Computer applications in stratigraphic analysis: John Wiley Sons, New York, 282 p.
Hempkins, W. B., 1965, The applicability of mathematical surfaces to correlation and prediction in Witwatersrand gold mines: Univ. Witwatersrand Info. Circ. No. 26, 26 p.
Hempkins, W. B., in press, A FORTRAN IV program for two–dimensional autocorrelation analysis of geologic and remotely–sensed data: Northwestern Univ. Report No. 21 (NGR–14–007–027), 54 p.
Huijbregts, C., and Matheron, G., 1970, Universal kriging (an optimal method for estimating and contouring in trend surface analysis): 9th Intern. sym. on decision-making in the mineral industries (proceedings to be published by Canadian Inst. Mining ), Montreal, preprint, 31 p.
Krige, D. G., 1962, Economic aspects of stoping through unpayable ore: Jour. South African Inst. Mining and Metallurgy, v. 63, p. 364–374.
Krige, D. G., 1966, Two-dimensional weighted moving average trend surfaces for ore valuation, in Sym. On mathematical statistics and computer applications in ore valuation: Jour. South African Inst. Mining and Metallurgy, p. 13–38.
Krige, D. G., 1966a, A study of gold and uranium distribution patterns in the Klerksdorp gold field: Geoexploration, v. 4, p. 43–53.
Krige, D. G., and Munro, A. H., 1968, A review of some conceptual and practical implications of the use of valuation surfaces for gold ore reserve estimation: Canadian Inst. Mining Metallurgy, Sp. Vol. 9, p. 33–40.
Krige, D. G., Watson, N. I., Oberholzer, W. J., and du Toit, S. R., 1969, The use of contour surfaces as predictive models for ore values: 8th Sym. on computer applications and operations research in the mineral industry, Salt Lake City, A.I.M.E., p. 127–161.
Krumbein, W. C., and Graybill, F. A., 1965, An introduction to statistical models in geology: McGraw-Hill Book Co., New York, 475 p.
Matern, B., 1960, Spatial variation: Medd. Skogsforskn Inst., v. 49, 144 p.
Matheron, G., 1962, Traite de geostatistique appliquee, tome 1: Bur. de Recherches Geol. et Min. Mem., no. 14, 333 p.
Matheron, G., 1965, Les variables regionalisees et leur estimation: Masson, Paris, 306 p.
Matheron, G., 1967, Kriging or polynomial interpolation procedures: Canadian Inst. Mining Metallurgy Bull., v. 70, p. 240–244.
Matheron, G., 1969, Cours de geostatistique: Cahiers du Centre Morph. Math. Fontainebleau, Ec. Nat. Sup. Mines Paris, v. 2, 82 p.
Papoulis, A., 1968, Systems and transforms with applications in optics: McGraw-Hill Book Co., New York, 474 p.
Parzen, E., 1967, Time series analysis for models of signal plus white noise, in Spectral analysis of time series: John Wiley Sons, New York, p. 233–257.
Pielou, E. C., 1964, The spatial pattern of two-phase patchworks of vegetation: Biometrics, v. 20, p. 156–167.
Salgado, J. A., 1968, Evaluacion de muestreos en minas de oro mediante computadores digitales: Escuela de Geol. Min. Met., Caracas, G.E.O.S., v. 18, p. 48–70.
Switzer, P., 1965, A random set process in the plane with a Markovian property: Ann. Math. Stat., v. 36, p. 1859–1863.
Vistelius, A. B., 1967, Basic features of porosity distribution in a typical section of Paleozoic rocks in the Volga Region, and the method of studying it, in Studies in mathematical geology (trans. from Russian ): Consultants Bureau, New York, p. 241–251.
Vistelius, A. B., 1969, Stochastic model of granitic rock crystallization and searching for hidden high temperature ore bodies: Proc. Sym. on mathematical methods in geology and geophysics, Pribram, Czechoslovakia, preprint, 6 p.
Watson, G. S., 1967, Linear least squares regression: Ann. Math. Stat., v. 38, p. 1679–1699.
Watson, G. S., 1969, Trend surf ae analysis and spatial correlation: Tech. Report 124, Dept. Statistics, Johns Hopkins Univ., 11 p.
Watson, M. I., 1968, Methods and models in ore evaluation: South African Chamber of Mines, Res. Rept. No. 55 /68, 28 p.
Whittle, P., 1954, On stationary processes in the plane: Biometrika, v. 41, p. 434–449.
Whittle, P., 1962, Topographic correlation, power-low covariance, functions and diffusion: Biometrika, v. 49, p. 305–314.
Whittle, P., 1963, Stochastic processes in several dimensions: 34th Sess. Intern. Stat. Inst. Bull., The Hague, p. 974–985.
Whittle, P., 1963a, Prediction and regulation: Van Nostrand Co., Princeton, N. J., 147 p.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1970 Plenum Press, New York
About this chapter
Cite this chapter
Agterberg, F.P. (1970). Autocorrelation Functions in Geology. In: Merriam, D.F. (eds) Geostatistics. Computer Applications in the Earth Sciences. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-7103-2_10
Download citation
DOI: https://doi.org/10.1007/978-1-4615-7103-2_10
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4615-7105-6
Online ISBN: 978-1-4615-7103-2
eBook Packages: Springer Book Archive