Abstract
Coregionalization analysis has been presented as a method of multi-scale analysis for multivariate spatial data. Despite an increasing use of this method in environmental and earth sciences, the uncertainty associated with the estimation of parameters in coregionalization analysis (e.g., sills and functions of sills) is potentially high and has not yet been characterized. This article aims to discuss the theory underlying coregionalization analysis and assess the robustness and limits of the method. A theoretical framework is developed to calculate the ergodic and fluctuation variance-covariance matrices of least-squares estimators of sills in the linear model of coregionalization. To adjust for the positive semidefiniteness constraint on estimated coregionalization matrices, a confidence interval estimation procedure for sills and functions of sills is presented. Thereafter, the relative importance of uncertainty measures (bias and variance) for sills and structural coefficients of correlation and determination is assessed under different scenarios to identify factors controlling their uncertainty. Our results show that the sampling grid density, the choice of the least-squares estimator of sills, the positive semidefiniteness constraint, the presence of scale dependence in the correlations, and the number and range of variogram models, all affect the level of uncertainty, sometimes through multiple interactions. The asymptotic properties of variogram model parameter estimators in a bounded sampling domain impose a theoretical limit to their accuracy and precision. Because of this limit, the uncertainty was found to be high for several scenarios, especially with three variogram models, and was often more dependent on the ratio of variogram range to domain extent than on the sampling grid density. In practice, in the coregionalization analysis of a real dataset, the circular requirement for sill estimates in the calculation of uncertainty measures makes the quantification of uncertainty very problematic, if not impossible. The use of coregionalization analysis must be made with due knowledge of the uncertainty levels and limits of the method.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bocchi, S., Castrignanò, A., Fornaro, F., and Maggiore, T., 2000, Application of factorial kriging for mapping soil variation at field scale: Eur. J. Agron., v. 13, no. 4, p. 295–308.
Bourennane, H., Salvador-Blanes, S., Cornu, S., and King, D., 2003, Scale of spatial dependence between chemical properties of topsoil and subsoil over a geologically contrasted area (Massif central, France): Geoderma, v. 112, no. 3–4, p. 235–251.
Castrignanò, A., Giugliarini, L., Risaliti, R., and Martinelli, N., 2000a, Study of spatial relationships among some soil physico-chemical properties of a field in central Italy using multivariate geostatistics: Geoderma, v. 97, no. 1–2, p. 39–60.
Castrignanò, A., Goovaerts, P., Lulli, L., and Bragato, G., 2000b, A geostatistical approach to estimate probability of occurrence of Tuber melanosporum in relation to some soil properties: Geoderma, v. 98, no. 3–4, p. 95–113.
Chilès, J.-P., and Delfiner, P., 1999, Geostatistics: Modeling Spatial Uncertainty: Wiley, New York, 695 p.
Cressie, N. A. C., 1993, Statistics for Spatial Data: Wiley, New York, 900 p.
Dobermann, A., Goovaerts, P., and Neue, H. U., 1997, Scale-dependent correlations among soil properties in two tropical lowland rice fields: Soil Sci. Soc. Am. J., v. 61, no. 5, p. 1483–1496.
Goovaerts, P., 1992, Factorial kriging analysis: A useful tool for exploring the structure of multivariate spatial soil information: J. Soil Sci., v. 43, p. 597–619.
Goovaerts, P., 1994, Study of spatial relationships between 2 sets of variables using multivariate geostatistics: Geoderma, v. 62, no. 1–3, p. 93–107.
Goovaerts, P., 1997, Geostatistics for Natural Resources Evaluation: Oxford University Press, New York, 483 p.
Goovaerts, P., and Webster, R., 1994, Scale-dependent correlation between topsoil copper and cobalt concentrations in Scotland: Eur. J. Soil Sci., v. 45, no. 1, p. 79–95.
Goovaerts, P., Sonnet, P., and Navarre, A., 1993, Factorial kriging analysis of springwater contents in the Dyle River basin, Belgium: Water Resour. Res., v. 29, no. 7, p. 2115–2125.
Goulard, M., 1989, Inference in a regionalization model, in Armstrong, M., ed., Geostatistics, vol. 1: Kluwer Academic, Dordrecht, pp. 397–408.
Goulard, M., and Voltz, M., 1992, Linear coregionalization model: Tools for estimation and choice of cross-variogram matrix: Math. Geol., v. 24, no. 3, p. 269–286.
Isaaks, E. H., and Srivastava, R. M., 1989, Applied Geostatistics: Oxford University Press, New York, 561 p.
Journel, A. G., and Huijbregts, C. J., 1978, Mining geostatistics: Academic, London, 600 p.
Larocque, G., Dutilleul, P., Pelletier, B., and Fyles, J. W., 2006, Conditional Gaussian co-simulation of regionalized components of soil variation: Geoderma, v. 134, no. 1, p. 1–16.
Lin, Y. P., 2002, Multivariate geostatistical methods to identify and map spatial variations of soil heavy metals: Environ. Geol., v. 42, p. 1–10.
Marchant, B. P., and Lark, R. M., 2004, Estimating variogram uncertainty: Math. Geol., v. 36, no. 8, p. 867–898.
Matheron, G., 1965, Les Variables Régionalisées et Leur Estimation, une Application de la Théorie des Fonctions Aléatoires aux Sciences de la Nature: Masson, Paris, 305 p.
Matheron, G., 1978, Estimer et Choisir: Un Essai sur la Pratique des Probabilités, Les Cahiers du Centre de Morphologie Mathématique, fasc. 7: Centre de Géostatistique, Fontainebleau, 175 p.
Matheron, G., 1982, Pour une analyse krigeante des données régionalisées: Rapport N-732, Centre de Géostatistique, Fontainebleau
Matheron, G., 1989, Estimating and Choosing: An Essay on Probability in Practice: Springer, Berlin, 141 p.
Müller, W. G., and Zimmerman, D. L., 1999, Optimal designs for variogram estimation: Environmetrics, v. 10, no. 1, p. 23–37.
Ortiz, C. J., and Deutsch, C. V., 2002, Calculation of uncertainty in the variogram: Math. Geol., v. 34, no. 2, p. 169–183.
Pardo-Igúzquiza, E., and Dowd, P., 2001, Variance-covariance matrix of the experimental variogram: Assessing variogram uncertainty: Math. Geol., v. 33, no. 4, p. 397–419.
Pardo-Igúzquiza, E., and Dowd, P., 2003, Assessment of the uncertainty of spatial covariance parameters of soil properties and its use in applications: Soil Sci., v. 168, no. 11, p. 769–781.
Pelletier, B., Dutilleul, P., Larocque, G., and Fyles, J. W., 2004, Fitting the linear model of coregionalization by generalized least squares: Math. Geol., v. 36, no. 3, p. 323–343.
Pettitt, A. N., and McBratney, A. B., 1993, Sampling designs for estimating spatial variance components: Appl. Stat. J. Roy. Stat. C, v. 42, no. 1, p. 185–209.
Searle, S. R., 1971, Linear Models: Wiley, New York, 532 p.
Vargas-Guzman, J. A., Warrick, A. W., and Myers, D. E., 2002, Coregionalization by linear combination of nonorthogonal components: Math. Geol., v. 34, no. 4, p. 401–415.
Wackernagel, H., 2003, Multivariate Geostatistics: An Introduction with Applications Springer, Berlin, 387 p.
Warrick, A. W., and Myers, D., 1987, Optimization of sampling locations for variogram calculations: Water Resour. Res., v. 23, no. 3, p. 496–500.
Webster, R., and Oliver, M. A., 1992, Sample adequately to estimate variograms of soil properties: J. Soil Sci., v. 43, no. 1, p. 177–192.
Webster, R., Atteia, O., and Dubois, J. P., 1994, Coregionalization of trace metals in the soil in the Swiss Jura: Eur. J. Soil Sci., v. 45, no. 2, p. 205–218.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Larocque, G., Dutilleul, P., Pelletier, B. et al. Characterization and Quantification of Uncertainty in Coregionalization Analysis. Math Geol 39, 263–288 (2007). https://doi.org/10.1007/s11004-007-9086-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11004-007-9086-8