Mathematical Geology

, Volume 28, Issue 1, pp 111–132 | Cite as

Trend removal in spatially correlated datasets

  • Anita Singh
  • Ashok K. Singh


Typically, datasets originated from mining exploration sites, industrially polluted and hazardous waste sites are correlated spatially over the region under investigation. Ordinary kriging (OK) is a well-established geostatistical tool used for predicting variables, such as precious metal contents, biomass, species counts, and environmental pollutants at unsampled spatial locations based on data collected from the neighboring sampled locations at these sites. One of the assumptions required to perform OK is that the mean of the characteristic of concern is constant for the entire region under consideration (e.g., there is no spatial trend present in the contaminant distribution across the site). This assumption may be violated by dalasets obtained from environmental applications. The occurrence of spatial trend in a dataset collected from a polluted site is an indication of the presence of two or more statistical populations (strata) with significantly different mean concentrations. Use of OK in these situations can result in inaccurate kriging estimates with higher SDs which, in turn, can lead to incorrect decisions regarding all subsequent environmental monitoring and remediation activities. A univariate and a multivariate approach have been described to identify spatial trend that may be present at the site. The trend then is removed by subtracting the respective means from the corresponding populations. The results of OK before and after trend removal are being compared. Using a real dataset, it is shown that standard deviations (SDs) of the kriging estimates obtained after trend removal are uniformly smaller than the corresponding SDs of the estimates obtained without the trend removal.

Key words

discriminant analysis kriging principal component analysis robust M-estimation separation of mixed populations spatial trend 


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  1. Campbell, N. A., 1984, Mixture models and atypical values: Math. Geology, v. 16, no. 5, p. 465–477.CrossRefGoogle Scholar
  2. Cressie, N., 1994, Statistics for spatial data (revised ed.): John Wily & Sons, New York, 900 p.Google Scholar
  3. Devlin, S. J., Gnanadesikan, R., and Kettenring, J. R., 1981, Robust estimation of dispersion matrices and principal components: Jour. Am. Statis. Assoc., v. 76, p. 354–362.Google Scholar
  4. Fleischhauer, H., and Korte, N., 1990, Formulation of cleanup standards for trace elements with probability plots: Environ. Management, v. 14, no. 1, p. 95–105.Google Scholar
  5. GEO-EAS 1.2.1., 1991, User's Guide. United States Environmental Protection Agency, EMSL, Las Vegas, NV, EPA 600/8-91/008.Google Scholar
  6. Gilles, B., and Marcotte, D., 1991, Multivariate variogram and its applications to linear model coregionalization: Math. Geology, v. 23, no. 7, p. 899–928.CrossRefGoogle Scholar
  7. Harff, J., and Davis, J. C., 1990. Regionalization in geology by multivariate classification: Math. Geology, v. 22, no. 5, p. 573–588.CrossRefGoogle Scholar
  8. Hopke, P. K., and Massart, D. L., 1993, Reference datasets for chemometrical methods testing: Chemometrics and Intelligent Laboratory Systems, v. 19, p. 35–41.CrossRefGoogle Scholar
  9. Jolliffe, I. T., 1986, Principal component analysis: Springer-Verlag, New York, 271 p.Google Scholar
  10. Journel, A. G., and Rossi, M. E., 1989, When do we need a trend model in kriging?: Math. Geology, v. 21, no. 7, p. 715–739.CrossRefGoogle Scholar
  11. Matheron, G., 1971, The theory of regionalized variables and its applications: Cahiers du Centre de Morphologie Mathematique no. 5, Fontainebleau, France, 211 p.Google Scholar
  12. Matheron, G., 1976, A simple substitute for conditional expectation: the disjunctive kriging,in Guarascio, M., David, M., and Huijbregts, C., eds., Advanced geostatistics in mining industry: D. Reidel Publ. Co., Dordrecht, The Netherlands, p. 221–236.Google Scholar
  13. Puente, C. E., and Bras, R. L., 1986, Disjunctive kriging, universal kriging, or no kriging: small sample results with simulated fields: Math. Geology, v. 18, no. 3, p. 287–305.CrossRefGoogle Scholar
  14. Rouhani, S., and Myers, D. E., 1990, Problems in space-time kriging of geohydrological data: Math. Geology, v. 22, no. 5, p. 611–623.CrossRefGoogle Scholar
  15. Seber, G. A. F., 1984, Multivariate observations: John Wiley & Sons, New York, 686 p.Google Scholar
  16. Singh, A., 1993, Omnibus robust procedures for assessment of multivariate normality and detection of multivariate outlier in multivariate environmental statistics,in Patil, G. P., and Rao, C. R., eds., Elsevier Sci. Publ., Amsterdam, v. 6, p. 445–488.Google Scholar
  17. Singh, A., Singh, A. K., and Flatman, G. T., 1994, Estimation of background levels of contaminants: Math. Geology, v. 26, no. 3, p. 361–388.CrossRefGoogle Scholar

Copyright information

© International Association for Mathematical Geology 1996

Authors and Affiliations

  • Anita Singh
    • 1
  • Ashok K. Singh
    • 2
  1. 1.Lockheed Environmental Systems & Technologies CompanyLas Vegas
  2. 2.Department of MathematicsUniversity of NevadaLas Vegas

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