Empirical variogram for achieving the best valid variogram

  • Mahdi, Esam (Department of Mathematics, Statistics, and Physics, Qatar University) ;
  • Abuzaid, Ali H. (Department of Mathematics, Al-Azhar University) ;
  • Atta, Abdu M.A. (Department of Mathematics, Statistics, and Physics, Qatar University)
  • Received : 2020.05.16
  • Accepted : 2020.08.17
  • Published : 2020.09.30


Modeling the statistical autocorrelations in spatial data is often achieved through the estimation of the variograms, where the selection of the appropriate valid variogram model, especially for small samples, is crucial for achieving precise spatial prediction results from kriging interpolations. To estimate such a variogram, we traditionally start by computing the empirical variogram (traditional Matheron or robust Cressie-Hawkins or kernel-based nonparametric approaches). In this article, we conduct numerical studies comparing the performance of these empirical variograms. In most situations, the nonparametric empirical variable nearest-neighbor (VNN) showed better performance than its competitors (Matheron, Cressie-Hawkins, and Nadaraya-Watson). The analysis of the spatial groundwater dataset used in this article suggests that the wave variogram model, with hole effect structure, fitted to the empirical VNN variogram is the most appropriate choice. This selected variogram is used with the ordinary kriging model to produce the predicted pollution map of the nitrate concentrations in groundwater dataset.


  1. Al-Najar H (2011). The integration of FAO-CropWat model and GIS techniques for estimating irrigation water requirement and its application in the Gaza strip, Natural Resources, 2, 146-154.
  2. Armstrong M and Jabin R (1981). Variogram models must be positive-definite, Journal of the International Association of Mathematical Geology, 13, 5-459.
  3. Barry RP and Pace RK (1997). Kriging with large data sets using sparse matrix techniques, Communications in Statistics - Simulation and Computation, 26, 619-629.
  4. Berman M and Diggle P (1989). Estimating weighted integrals of the second-Order Intensity of a spatial point process, Journal of the Royal Statistical Society: Series B, 51, 81-92.
  5. Chiles JP and Delfiner P (1999). Geostatistics: Modelling Spatial Uncertainty, John Wiley & Sons, Wiley-Interscience.
  6. Cressie N (1985). Fitting variogram models by weighted least squares, Mathematical Geology, 17, 563-586.
  7. Cressie N (1993). Statistics for Spatial Data, John Wiley & Sons, New York.
  8. Cressie N and Hawkins DM (1980). Robust estimation of the variogram, Journal of the International Association of Mathematical Geology, 12, 115-125.
  9. Croux C and Rousseeuw PJ (1992). Time-efficient algorithms for two highly robust estimators of scale, Computational Statistics, 1, 411-428.
  10. Diggle PJ (1985). A kernel method for smoothing point process data, Applied Statistics, 34, 138-147.
  11. Diggle PJ, Gates DJ, and Stibbard A (1987). A nonparametric estimator for pairwise-interaction point processes, Biometrika, 74, 763-770.
  12. Dunn MR (1983). A simple sufficient condition for a variogram model to yield positive varianees under restrictions, Mathematical Geology, 15, 553-564.
  13. European Union (December 2006). Directive 2006/118/EC of the European parliament and of the council on the protection of groundwater against pollution and deterioration, Official Journal of the European Union, L 372, 19-31.
  14. Fernandez-Casal R (2016). npsp: Nonparametric Spatial (Geo)statistics, R package version 0.5-3.
  15. Garcia-Soidan PH, Febrero-Bande M, and Gonzalez-Manteiga W (2004). Nonparametric kernel estimation of an isotropic variogram, Journal of Statistical Planning and Inference, 121, 65-92.
  16. Garcia-Soidan PH, Gonzalez-Manteiga W, and Febrero-Bande M (2003). Local linear regression estimation of the variogram, Statistics and Probability Letters, 64, 169-179.
  17. Garcia-Soidan PH and Menezes R (2012). Estimation of the spatial distribution through the kernel indicator variogram, Environmetrics, 23, 535-548.
  18. Genton MG (1998). Highly robust variogram estimation, Mathematical Geology, 30, 213-221.
  19. Gorsich DJ and Genton MG (2000). Variogram model selection via nonparametric derivative estimation, International Association for Mathematical Geology, 32, 249-270.
  20. Graler B, Pebesma E, and Heuvelink G (2016). Spatio-Temporal Interpolation using gstat, The R Journal, 8, 204-218.
  21. Hall P, Fisher NI, and Hoffmann B (1994). On the nonparametric estimation of covariance functions, The Annals of Statistics, 22, 2115-2134.
  22. Hastie T, Tibshirani R, and Friedman J (2009). The Elements of Statistical Learning, Springer, New York.
  23. Huang C, Hsing T, and Cressie N (2011). Nonparametric estimation of the variogram and its spectrum, Biometrika, 98, 775-789.
  24. Isaaks EH and Srivastava RM (1989). An Introduction to Applied Geostatistics, Oxford University Press, New York.
  25. Jammalamadaka SR and Sengupta A (2001). Topics in Circular Statistics, World Scientific, Singapore.
  26. Jin R and Kelly GE (2017). A comparison of sampling grids, cut-off distance and type of residuals in parametric variogram estimation, Communications in Statistics - Simulation and Computation, 46, 1781-1795.
  27. Lahiri SN, Kaiser MS, Cressie N, and Hsu NJ (1999). Prediction of spatial cumulative distribution functions using subsampling, Journal of the American Statistical Association, 94, 86-97.
  28. Matern B (1960; reprinted 1986). Spatial Variation (2nd ed), Springer-Verlag, Berlin.
  29. Matheron G (1962). Trait'e de g'eostatistique appliqu'ee, tome i, memoires du Bureau de recherches geologiques et minieres, Editions Technip, 14, Paris.
  30. Matheron G (1963a). Trait'e de g'eostatistique appliqu'ee, tome ii, le krigeage. memoires du bureau de recherches geologiques et minieres, Editions Bureau de Recherches Geologiques et Minieres, 24, Paris.
  31. Matheron G (1963b). Principles of geostatistics, Economic Geology, 58, 1246-1266.
  32. Menezes R, Garcia-Soidan PH, and Febrero-Bande M (2005). A comparison of approaches for valid variogram achievement, Computational Statistics, 20, 623-642.
  33. Menezes R, Garcia-Soidan PH, and Febrero-Bande M (2008). A Kernel variogram estimator for clustered data, Scandinavian Journal of Statistics, 35, 18-37.
  34. Moran PAP (1950). Notes on continuous stochastic phenomena, Biometrika, 37, 17-23.
  35. Nolan BT, Ruddy BC, Hitt KJ, and Helsel DR (1998). A national look at nitrate contamination of ground water, Water Conditioning and Purification, 39, 76-79.
  36. Palestinian Central Bureau of Statistics (2014). Palestinian Central Bureau of Statistics reports. Available from:
  37. Pebesma EJ and Bivand RS (2005). Classes and methods for spatial data in R, R News, 5, 9-13.
  38. Ribeiro Jr PJ and Diggle PJ (2001). geoR: a package for geostatistical analysis, R-NEWS, 1, 15-18.
  39. Rousseeuw PJ and Croux C (1993). Alternatives to the median absolute deviation, Journal of the American Statistical Association, 88, 1273-1283.
  40. Shapiro A and Botha JD (1991). Variogram fitting with a general class of conditionally nonnegative definite functions, Computational Statistics and Data Analysis, 11, 11-96.
  41. Shomer B, Muller G, and Yahya A (2004). Potential use of treated wastewater and sludge in agricultural sector of the Gaza strip, Clean Technologies and Environmental Policy, 6, 128-137.
  42. Spalding RF and Exner ME (1993). Occurrence of nitrate in groundwater - a review, Journal of Environmental Quality, 22, 392-402.
  43. Stein ML (1999). Interpolation of Spatial Data - Some Theory for Kriging, Springer Verlag, New York.
  44. United Nations (August 2012). Gaza in 2020: a liveable place?, United Nations Relief and Works Agency for Palestine Refugees in the Near East (UNRWA), 1-24.
  45. United States Environmental Protection Agency (1995). Drinking water regulations and health advisories, Office of Water, Washington.
  46. Waller LA and Crawford CAG (2004). Applied Spatial Statistics for Public Health Data, John Wiley & Sons, Hoboken, NJ.
  47. Yaglom A (1987). Correlation Theory of Stationary and Related Random Functions, Springer Verlag, New York.
  48. Yu K, Mateu J, and Porcu E (2007). A kernel-based method for nonparametric estimation of variograms, Statistica Neerlandica, 61, 173-197.