Abstract
Block-sample covariances may be calculated by discretizing a block into regularly spaced grid points, computing punctual covariance between each grid point and the sample, then averaging. Gauss quadrature is a better, more accurate method for calculating block-sample covariance as has been demonstrated in the past by other authors (the history of which is reviewed herein). This prior research is expanded upon to provide considerably more detail on Gauss quadrature for approximating the areal or volumetric integral for block-sample covariance. A 4 × 4 Gauss point rule is shown to be optimal for this procedure. Moreover, pseudo-computer algorithms are presented to show how to implement Gauss quadrature in existing computer programs which perform block kriging.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramowitz, M., and Stegun, I. A., Handbook of Mathematical Functions: Dover, New York, 1046 p.
Carr, J. R., 1992, Numerical Analysis for the Geological Sciences: Macmillan, New York (manuscript in preparation).
Clark, I., 1976a, Some Auxiliary Functions for the Spherical Model of Geostatistics: Comput. Geosci., v. 1, p. 255–263.
Clark, I., 1976b, Some Practical Computational Aspects of Mine Planning,in M. Guarascio, M. David, and C. Huijbregts (Eds.), Advanced Geostatistics in the Mining Industry: Reidel, Dordrecht, p. 391–402.
Clark, I., 1977, Practical Kriging in Three Dimensions: Comput. Geosci., v. 3, p. 173–180.
Cook, R. D., 1974, Concepts and Applications of Finite Element Analysis: Wiley, New York, 402 p.
David, M., 1977, Geostatistical Ore Reserve Estimation: Elsevier, Amsterdam, 364 p.
Davis, M. W. D., and David, M., 1978, The Numerical Calculation of Integrals of an Isotropic Variogram Function on Hypercubes: Math. Geol., v. 10, p. 311–314.
Journel, A. G., and Huijbregts, Ch. J., 1978, Mining Geostatistics: Academic Press, London, 60 p.
Krylov, V. I., 1962, Approximate Calculation of Integrals: Macmillan, New York, 357 p.
Marechal, A., 1976, Calcul Numerique des Variogrammes Regularises: Internal Report N-467, CG, Fontainebleau.
Scarborough, J. B., 1955, Numerical Mathematical Analysis, 3rd ed.: The Johns Hopkins Press, Baltimore, 554 p.
Zhang, R., Warrick, A. W., and Myers, D. E., 1990, Variance as a Function of Sample Support Size: Math. Geol., v. 22, p. 107–122.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Carr, J.R., Palmer, J.A. Revisiting the accurate calculation of block-sample covariances using Gauss quadrature. Math Geol 25, 507–524 (1993). https://doi.org/10.1007/BF00890243
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00890243