Abstract
Before optimal linear prediction can be performed on spatial data sets, the variogram is usually estimated at various lags and a parametric model is fitted to those estimates. Apart from possible a priori knowledge about the process and the user's subjectivity, there is no standard methodology for choosing among valid variogram models like the spherical or the exponential ones. This paper discusses the nonparametric estimation of the variogram and its derivative, based on the spectral representation of positive definite functions. The use of the estimated derivative to help choose among valid parametric variogram models is presented. Once a model is selected, its parameters can be estimated—for example, by generalized least squares. A small simulation study is performed that demonstrates the usefulness of estimating the derivative to help model selection and illustrates the issue of aliasing. MATLAB software for nonparametric variogram derivative estimation is available at http://www-math.mit.edu/~gorsich/derivative.html. An application to the Walker Lake data set is also presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Arfken, G., 1985, Mathematical methods for physicists: Academic Press, New York, 985 p.
Barry, R. P., and Ver Hoef, J. M., 1996, Blackbox kriging: spatial prediction without specifying variogram models: Journal of Agricultural, Biological and Environmental Statistics, v. 1, no. 3, p. 297–322.
Bochner, S., 1955, Harmonic analysis and the theory of probability: University of California Press, Berkley and Los Angeles, 176 p.
Cherry, S., 1997, Non-parametric estimation of the sill in geostatistics: Environmetrics, v. 8, p. 13–27.
Cherry, S., Banfield, J., and Quimby, W. F., 1996, An evaluation of a non-parametric method of estimating semi-variograms of isotropic spatial processes: Journal of Applied Statistics, v. 23, no. 4, p. 435–449.
Clark, I., 1979, Practical geostatistics: Applied Science Publishers, Essex, England, 129 p.
Cressie, N., 1985, Fitting variogram models by weighted least squares: Math. Geology, v. 17, p. 563–586.
Cressie, N., 1993, Statistics for spatial data: John Wiley & Sons, New York, 900 p.
Ecker, M. D., and Gelfand, A. E., 1997, Bayesian variogram modeling for an isotropic spatial process: Journal of Agricultural, Biological and Environmental Statistics, v. 2, no. 4, p. 347–369.
Genton, M. G., 1998a, Highly robust variogram estimation: Math. Geology, v. 30, no. 2, p. 213–221.
Genton, M. G., 1998b, Variogram fitting by generalized least squares using an explicit formula for the covariance structure: Math. Geology, v. 30, no. 4, p. 323–345.
Härdle, W., 1989, Applied nonparametric regression: Cambridge University Press, 333 p.
Isaaks, E. H., and Srivastava, R. M., 1989, An introduction to applied geostatistics: Oxford University Press, 561 p.
Journel, A. G., and Huijbregts, Ch. J., 1978, Mining geostatistics: Academic Press, London, 600 p.
Matheron, G., 1962, Traité de géostatistique appliquée, Tome I: Mémoires du Bureau de Recherches Géologiques et Minières, no. 14, Editions Technip, Paris, 333 p.
Oppenheim, A. V., and Schafer, R. W., 1989. Discrete-time signal processing: Prentice-Hall Signal Processing Series, Englewood Cliffs, New Jersey, 879 p.
Schoenberg, I. J., 1938, Metric spaces and completely monotone functions: Annals of Mathematics, v. 39, no. 4, p. 811–841.
Shapiro, A., and Botha, J. D., 1991. Variogram fitting with a general class of conditionally nonnegative definite functions: Computational Statistics and Data Analysis, v. 11, p. 87–96.
Stein, M., 1988, Asymptotically efficient prediction of a random field with a misspecified covariance function: Annals of Statistics, v. 16, no. 1, p. 55–63.
Stein, M., 1990, Uniform asymptotic optimality of linear predictions of a random field using an incorrect second-order structure: Annals of Statistics, v. 18, no. 2, p. 850–872.
Wand, M. P., and Jones, M. C., 1995, Kernel smoothing: Chapman & Hall, 212 p.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Gorsich, D.J., Genton, M.G. Variogram Model Selection via Nonparametric Derivative Estimation. Mathematical Geology 32, 249–270 (2000). https://doi.org/10.1023/A:1007563809463
Issue Date:
DOI: https://doi.org/10.1023/A:1007563809463