Abstract
The problem of estimating a regionalized variable in the presence of other secondary variables is encountered in spatial investigations. Given a context in which the secondary variable is known everywhere (or can be estimated with great precision), different estimation methods are compared: regression, regression with residual simple kriging, kriging, simple kriging with a mean obtained by regression, kriging with an external drift, and cokriging. The study focuses on 19 pairs of regionalized variables from five different datasets representing different domains (geochemical, environmental, geotechnical). The methods are compared by cross-validation using the mean absolute error as criterion. For correlations between the principal and secondary variable under 0.4, similar results are obtained using kriging and cokriging, and these methods are superior slightly to the other approaches in terms of minimizing estimation error. For correlations greater than 0.4, cokriging generally performs better than other methods, with a reduction in mean absolute errors that can reach 46% when there is a high degree of correlation between the variables. Kriging with an external drift or kriging the residuals of a regression (SKR) are almost as precise as cokriging.
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Ahmed, S., and de Marsily, G., 1987, Comparison of geostatistical methods for estimating transmissivity using data on transmissivity and specific capcity: Water Resources Research, v. 23, no. 9, p. 1717–1737.
Beaumier, M., 1987, Géochimie des sédiments de lac de la région de Schefferville: MB 87-32. M.E.R.Q., 382 p.
Castelier, E., 1993, Dérive externe et régression linéaire: Compte-rendu des journées de géostatistique (25–26 mai 1993), Fontainebleau: Cahiers de géostatistique, Fascicule 3, p. 47–59.
Creutin, J. D., Andrieu, H., and Delerieu, G., 1989, Une simplification du cokrigeage appliquée à l'étalonnage d'images de télédétection en hydrométéorologie,in Armstrong, M., ed., Geostatistics (Vol. 2): Kluwer Acad. Publ., Dordrect, p. 761–772.
David, M., 1977, Geostatistical ore reserve estimation: Elsevier Sci. Publ. Co., New York, 364 p.
David, M., Marcotte, D., and Soulié, M., 1984, Conditional bias in kriging and a suggested correction,in Verly, G., and others, ed., Geostatistics for natural resources characterisation: Reidel Publ. Co., Dordrecht, p. 217–230.
Delerieu, G., Bellon, A., and Creutin, J. D., 1988, Estimation de lames d'eau spatiales à l'aide de données de pluviométrie et de radar météorologique: Applications au pas de temps journalier dans la région de Montréal: Jour. Hydrol., v. 98. no. 3–4, p. 315–344.
Delfiner, P., Delhomme, J. P., and Pelissier Combescure, J., 1983, Application of geostatistical analysis to the evaluation of petroleum reservoirs with well logs: 24th Annual Login Symposium (Calgary, Canada), v. 24, no. 1, p. WW1-WW26.
Delhomme, J. P., 1979, Étude de la géométrie du réservoir de Chemery: Internal Report, Centre d'Informatique Géologique. École des Mines de Paris, Fontainebleau, Rept. LHM/RD/79/41, unpaginated.
Deutsch, C., 1991, External drift- the inside story: Geostatistics (an interdisciplinary geostatistics) Newsletter, v. 5, p. 2.
Draper, N. R., and Smith, H., 1981, Applied regression analysis (2nd ed.): John Wiley & Sons, New York, 709 p.
Englund, E., and Sparks, A., 1988, GEO-EAS (Geostalistical Environmental Assessment Software), User's Guide: Environmental Monitoring Systems Laboratory, Office of Research and Development, U.S. Environmental Protection Agency, EPA 4-88 033, 196 p.
Galli, A., and Meunier, G. 1987, Study of a gas reservoir using the external drift method,in Matheron, G., and Armstrong, M., eds., Geostatistical case studies. D. Reidel Publ. Co., Dordrecht, p. 105–119.
Gallichand, J., Buckland, G. D., Marcotte, D., and Hendry, M. J., 1992, Spatial interpolation of soil salinity and sodicity for a saline soil in southern Alberta: Can. Jour. Soil Sci., v. 72, no. 4. p. 503–516.
Grunsky, E. C., 1986a, Recognition of alteration in volcanic rocks using statistical analysis of lithogeochemical data: Jour. Geochemical Exploration, v. 25, no. 1–2, p. 157–183.
Grunsky, E. C., 1986b, Recognition of alteration in volcanic rocks using statistical analysis of lithogeochemical data,in Wood, J., and Wallace, H., eds., Volcanology and mineral exploration: Ontario Geol. Survey, Misc. Papers, p. 124–173.
Istok, J. D., Smyth, J. D., and Flin, A. L., 1993, Multivariate geostatistical analysis of groundwater contamination: a case history: Ground Water, v. 31, no. 1, p. 63–74.
Jensen, L. S., 1975, Geology of Clifford and Ben Nevis townships, District of Cochrane, Ontario: Div. Mines, GR132, 55 p. (accompanied by Map 2283).
Journel, A. G., 1984, MAD and conditional quantiles estimators,in Verly, G., ed., Proc. NATO ASI, geostatistics for natural resources characterisation: Reidel, Dordecht, Holland, p. 261–270.
Journel, A. G., and Huijbregts, C. J., 1978, Mining geostatistics: Academic Press, London, 600 p.
Kirsch, Ch., and Pawlowsky, V., 1985, Modélisation de variogrammes croisés—comment satisfaire à l'inégalité de Cauchy-Schwartz: Sciences de la Terre, v. 24, p. 53–62.
Maréchal, A., 1984, Kriging seismic data in presence of faults,in Verly, G., ed., Proc. NATO ASI, geostatistics for natural resources characterisation: Reidel, Dordrecht, Holland, p. 271–294.
Marcotte, D., 1991, Cokriging with matlab: Computers & Geosciences, v. 17. no. 9, p. 1265–1280.
Matheron, G., 1965, Les variables régionalisées et leur estimation, une application des fonctions aléatoires aux sciences de la nature: Masson, Paris, 1305 p.
Matheron, G., 1970, La théorie des variables régionalisées et ses applications: Fascicule No. 5, Cahier du Centre de Morphologie Mathématique de Fontainbleau, 212 p.
Matheron, G. 1979, Recherche de simplification dans un problème de cokrigeage: Publ. N-628, Centre de Géostatistique. École des Mines de Paris, Fontainebleau, 19 p.
Morton, D. M., Baird, A. K., and Baird, K. W., 1969, The Lakeview Mountains Pluton, Southern California Batholith, II. Chemical composition and variation: Geol. Soc. America Bull., v. 80, no. 8, p. 1553–1564.
Myers, D., 1982, Matrix formulation of co-kriging: Math. Geology, v. 14, no. 3, p. 249–257.
Ouellet, J., 1985, Répartition spatiale des propriétés mécaniques des roches: Mémoire de Maîtrise es Sciences Appliquées, École Polytechnique de Montréal, 172 p.
Stein, A., Dooremolen, W. V., Bouma, J., and Bregt, A. K., 1988, Cokriging point data on moisture deficit: Soil Sci. Soc. Am. Jour., v. 52, no. 5, p. 1418–1423.
Vauclin, M., Vieira, S. R., Vachaud, G., and Nielsen, D. R., 1983, The use of cokriging with limited field soil observations: Soil Sci. Soc. Am. Jour., v. 47, no. 2, p. 175–184.
Wackernagel, H., 1994, Cokriging versus kriging in regionalized multivariate data analysis: Geo-Derma, v. 62, no. 1–3, p. 83–92.
Wolfe, W. J., 1977, Geochemical exploration of early precambrian sulphide mineralization in Ben Nevis Township, District of Cochrane: Ontario Geol. Survey, Study 19, 39 p.
Yates, M. V., and Yates, S. R., 1987, A comparison of geostatistical methods for estimating virus inactivation rates in ground water: Water Res., v. 21, no. 9, p. 1119–1125.
Yates, S. R., and Warrick, A. W., 1987, Estimating soil water content using cokriging: Soil Sci. Soc. Am. Jour., v. 51, no. 1, p. 23–30.
Zhang, R., Warrick, A. W., and Myers, D. E., 1992. Improvement of the prediction of soil particle size fractions using spectral properties: Geoderma, v. 52, no. 3–4, p. 223–234.
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Asli, M., Marcotte, D. Comparison of approaches to spatial estimation in a bivariate context. Math Geol 27, 641–658 (1995). https://doi.org/10.1007/BF02093905
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DOI: https://doi.org/10.1007/BF02093905