Abstract
In this article, we present the multivariable variogram, which is defined in a way similar to that of the traditional variogram, by the expected value of a distance, squared, in a space withp dimensions. Combined with the linear model of coregionalization, this tool provides a way for finding the elementary variograms that characterize the different spatial scales contained in a set of data withp variables. In the case in which the number of elementary components is less than or equal to the number of variables, it is possible, by means of nonlinear regression of variograms and cross-variograms, to estimate the coregionalization parameters directly in order to obtain the elementary variables themselves, either by cokriging or by direct matrix inversion. This new tool greatly simplifies the procedure proposed by Matheron (1982) and Wackernagel (1985). The search for the elementary variograms is carried out using only one variogram (multivariable), as opposed to thep(p + 1)/2 required by the Matheron approach. Direct estimation of the linear coregionalization model parameters involves the creation of semipositive definite coregionalization matrices of rank 1.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Baird, A. K., McIntyre, D. B., Welday, E. E., and Morton, D. M., 1967, A Test of Chemical Variability and Field Sampling Methods, Lakeview Mountain Tonalite, Lakeview Mountains Southern California Batholith: Calif. Div. Mines and Geology Spec. Rep. 92, p. 11–19.
Bourgault, G., 1990, Estimation et Filtrage Multidimensionnel de Variables Aléatoires Régionalisées: Ecole Polytechnique de Montréal, rapport préthèse, 75 p.
Daly, C., Jeulin, D., and Lajaunie, C., 1989, Application of Multivariate Kriging to the Processing of Noisy Images,in M. Armstrong (Ed.), Geostatistics, Vol. 2, Quantitative Geology and Geostatistics: Kluwer Academic Publishers, p. 749–760.
David, M., and Dagbert, M., 1974, Lakeview Revisited: Variograms and Correspondence Analysis—New Tools for the Understanding of Geochemical Data: Proc. 5th Exploration Geochemistry Symposium, Elsevier, Amsterdam, p. 163–181.
Davis, B. M., and Greenes, K. A., 1983, Estimation Using Spatially Distributed Multivariate Data: An Example With Coal Quality: Math. Geol., v. 15, p. 287–300.
Galli, A., and Wackernagel, H., 1987, Multivariate Geostatistical Methods for Spatial Data Analysis,in E. Diday et al. (Eds.), Data Analysis and Information, Vol. 5: North-Holland, Amsterdam.
Harff, J., and Davis, J. C., 1990, Regionalization in Geology by Multivariate Classification: Math. Geol., v. 22, p. 573–588.
Ibanez, F., 1976, Contribution à l'Analyse Mathématique des Evènements en Ecologie Planctonique: Bull. Inst. Ocea., Monaco, v. 72, 96 p.
Journel, A. G., 1988, New Distance Measures: The Route Toward Truly Non-Gaussian Geostatistics: Math. Geol., v. 20, p. 459–475.
Journel, A. G., and Huijbregts, Ch. J., 1978, Mining Geostatistics: Academic Press, London, 600 p.
Mackas, D. L., 1984, Spatial Autocorrelation of Plankton Community Composition in a Continental Shelf Ecosystem: Limn. and Ocean., v. 29, p. 451–471.
Mantel, N., 1967, The Detection of Disease Clustering and a Generalized Regression Approach: Cancer Res., v. 27, p. 209–220.
Marcotte, D., 1990, Lake Sediments in the Manicouagan Area: Multivariate Analysis and Variography Used to Enhance Anomalous Response: Statistical Applications in the Earth Sciences, GSC paper 89-9.
Marcotte, D., and Fox, J. S., 1990, The Schefferville Area: Multivariate Analysis and Variography Used to Enhance Interpretation of Lake Sediment Geochemical Data: Journal of Geochemical Exploration, v. 38, p. 247–263.
Marquardt, D., 1963, An Algorithm for Least-Squares Estimation of Nonlinear Parameter: SIAM Journal on Applied Mathematics, v. 11, p. 431–441.
Matheron, G., 1982, Pour une Analyse Krigeante des Données Régionalisées, N-732: Centre de Géostatistique et de Morphologie Mathématique, Fontainebleau, 22 p.
Moore, E. H., 1920, On the Reciprocal of the General Algebraic Matrix: Bull. Am. Math. Soc., v. 26, p. 394–395.
Myers, D., 1982, Matrix Formulation of Co-Kriging: Math. Geol., v. 14, p. 249–257.
Myers, D. E., 1983, Estimation of Linear Combinations and Cokriging: Math. Geol., v. 15, p. 633–637.
Myers, D. E., 1984, Cokriging: New Developments, in Geostatistics for Natural Resource Characterization, G. Verly et al. (Eds.): D. Reidel, Dordrecht.
Myers, D. E., 1985, Co-kriging: Methods and Alternatives,in P. Glaeser (Ed.), The Role of Data in Scientific Progress: Elsevier.
Myers, D. E., 1988a, Some Aspects of Multivariate Analysis,in C. F. Chung et al. (Eds.), Quantitative Analysis of Mineral and Energy Resources; Ridel, Dordrecht, p. 669–687.
Myers, D. E., 1988b, Multivariate Geostatistics for Environmental Monitoring: Sciences de la Terre, v. 27, p. 411–427.
Myers, D. E., 1988c, Interpolation with Positive Definite Functions: Sciences de la Terre, v. 28, p. 251–265.
Myers, D. E., and Carr, J. R., 1984, Co-kriging and Principal Components Analysis: Bentonite Data Revisited: Sciences de la Terre, v. 21, p. 65–77.
Oden, N. L., and Sokal, R. R., 1986, Directional Autocorrelation: An Extension of Spatial Correlograms to Two Dimensions: Syst. Zool., v. 35, p. 608–617.
Orlóci, L., 1967, An Agglomerative Method for Classification of Plant Communities: Journ. Ecol., v. 55, p. 193–206.
Penrose, R., 1955, A Generalized Inverse for Matrices: Proc. Cambridge Philos. Soc., v. 51, p. 406–418.
Sandjivy, L., 1984, The Factorial Kriging Analysis of Regionalized Data—Its Application to Geochemical Prospecting,in G. Verly et al. (Eds.), Geostatistics for Natural Resources Characterization: NATO-ASI, Ser. C, v. 122, Part 1, p. 559–572.
Serra, J., 1968, Les Structures Gigognes: Morphologie Mathématique et Interprétation Métallogénique: Mineralium Deposita, v. 3, p. 135–154.
Sokal, R. R., 1986, Spatial Data Analysis and Historical Processes,in E. Diday et al. (Eds.), Data Analysis and Informatics, IV: Proc. Fourth Int. Symp. Data Anal. Informatics, Versailles, France, 1985, North-Holland, Amsterdam, p. 29–43.
Wackernagel, H., 1985, L'Inférence d'un Modèle Linéaire en Géostatistique Multivariable: Thèse, Ecole Nationale Supérieure des Mines de Paris, 100 p.
Wackernagel, H., 1988, Geostatistical Techniques for Interpreting Multivariate Spatial Information,in C. F. Chung et al. (Eds.), Quantitative Analysis of Mineral Energy Resources: Reidel, Dordrecht, p. 393–409.
Wackernagel, H., 1989, Description of a Computer Program for Analysing Multivariate Spatially Distributed Data: Compt. and Geosc., v. 15, p. 593–598.
Wackernagel, H., and Butenuth, C., 1989, Caractérisation d'anomalies Géochimiques par la Géostatistique Multivariable: Journ. of Geochm. Explor., v. 32, p. 437–444.
Wackernagel, H., Petitgas, P., and Touffait, Y., 1989, Overview of Methods for Co-regionalization Analysis,in M. Armstrong (Ed.), Geostatistics, Vol. 1, Quantitative Geology and Geostatistics: Kluwer Academic Publishers, p. 409–420.
Webster, R., 1973, Automatic Soil-Boundary Location from Transect Data: Math. Geol., v. 5, p. 27–37.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bourgault, G., Marcotte, D. Multivariable variogram and its application to the linear model of coregionalization. Math Geol 23, 899–928 (1991). https://doi.org/10.1007/BF02066732
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02066732