Environmental Monitoring and Assessment

, Volume 185, Issue 1, pp 1–19 | Cite as

Comparison of stochastic and deterministic methods for mapping groundwater level spatial variability in sparsely monitored basins

  • Ε. A. Varouchakis
  • D. T. Hristopulos


In sparsely monitored basins, accurate mapping of the spatial variability of groundwater level requires the interpolation of scattered data. This paper presents a comparison of deterministic interpolation methods, i.e. inverse distance weight (IDW) and minimum curvature (MC), with stochastic methods, i.e. ordinary kriging (OK), universal kriging (UK) and kriging with Delaunay triangulation (DK). The study area is the Mires Basin of Mesara Valley in Crete (Greece). This sparsely sampled basin has limited groundwater resources which are vital for the island’s economy; spatial variations of the groundwater level are important for developing management and monitoring strategies. We evaluate the performance of the interpolation methods with respect to different statistical measures. The Spartan variogram family is applied for the first time to hydrological data and is shown to be optimal with respect to stochastic interpolation of this dataset. The three stochastic methods (OK, DK and UK) perform overall better than the deterministic counterparts (IDW and MC). DK, which is herein for the first time applied to hydrological data, yields the most accurate cross-validation estimate for the lowest value in the dataset. OK and UK lead to smooth isolevel contours, whilst DK and IDW generate more edges. The stochastic methods deliver estimates of prediction uncertainty which becomes highest near the southeastern border of the basin.


Environmental monitoring Geostatistics Kriging Groundwater level Ungauged basin Fractional Brownian motion 



We would like to thank Professor George Karatzas, Department of Environmental Engineering, Technical University of Crete, Greece, for many helpful discussions.


  1. Abedini, M. J., Nasseri, M., & Ansari, A. (2008). Cluster-based ordinary kriging of piezometric head in West Texas/New Mexico—Testing of hypothesis. Journal of Hydrology, 351(3–4), 360–367.CrossRefGoogle Scholar
  2. Aboufirassi, M., & Marino, M. (1983). Kriging of water levels in the Souss aquifer, Morocco. Mathematical Geology, 15(4), 537–551.CrossRefGoogle Scholar
  3. Ahmadi, S., & Sedghamiz, A. (2007). Geostatistical analysis of spatial and temporal variations of groundwater level. Environmental Monitoring and Assessment, 129(1), 277–294.CrossRefGoogle Scholar
  4. Ahmadi, S., & Sedghamiz, A. (2008). Application and evaluation of kriging and cokriging methods on groundwater depth mapping. Environmental Monitoring and Assessment, 138(1), 357–368.CrossRefGoogle Scholar
  5. Ahmed, S. (2007). Application of geostatistics in hydrosciences. In M. Thangarajan (Ed.), Groundwater (pp. 78–111). Dordrecht: Springer.CrossRefGoogle Scholar
  6. Briggs, I. C. (1974). Machine contouring using minimum curvature. Geophysics, 39(1), 39–48.CrossRefGoogle Scholar
  7. Brus, D. J., & Heuvelink, G. B. M. (2007). Optimization of sample patterns for universal kriging of environmental variables. Geoderma, 138(1–2), 86–95.CrossRefGoogle Scholar
  8. Buchanan, S., & Triantafilis, J. (2009). Mapping water table depth using geophysical and environmental variables. Ground Water, 47(1), 80–96.CrossRefGoogle Scholar
  9. Christakos, G. (1991). Random field models in earth sciences. San Diego: Academic.Google Scholar
  10. Cooke, R., Mostaghimi, S., & Parker, J. C. (1993). Estimating oil spill characteristics from oil heads in scattered monitoring wells. Environmental Monitoring and Assessment, 28(1), 33–51.CrossRefGoogle Scholar
  11. Cornford, D. (2005). Are comparative studies a waste of time? SIC2004. In G. Dubois (Ed.), EUR, 2005. Automatic mapping algorithms for routine and emergency monitoring data. EUR 21595 EN—Scientific and Technical Research Series (pp. 61–70). Luxembourg: Office for Official Publications of the European Communities. ISBN 92-894-9400-X.Google Scholar
  12. Cressie, N. (1993). Statistics for spatial data (revised edition). New York: Wiley.Google Scholar
  13. Dash, J., Sarangi, A., & Singh, D. (2010). Spatial variability of groundwater depth and quality parameters in the national capital territory of Delhi. Environmental Management, 45(3), 640–650.CrossRefGoogle Scholar
  14. Delhomme, J. P. (1974). La cartographie d’une grandeur physique a partir des donnees de differentes qualities. Proceedings of IAH Congress, Montpelier, France, Tome X (Part-1), pp. 185–194.Google Scholar
  15. Delhomme, J. P. (1978). Kriging in the hydrosciences. Advances in Water Resources, 1(5), 251–266.CrossRefGoogle Scholar
  16. Desbarats, A. J., Logan, C. E., Hinton, M. J., & Sharpe, D. R. (2002). On the kriging of water table elevations using collateral information from a digital elevation model. Journal of Hydrology, 255(1–4), 25–38.CrossRefGoogle Scholar
  17. Deutsch, C. V., & Journel, A. G. (1992). GSLIB. Geostatistical software library and user’s guide. New York: Oxford University Press.Google Scholar
  18. Donta, A. A., Lange, M. A. & Herrmann, A. (2006). Water on Mediterranean islands: Current conditions and prospects for sustainable management. Project No. EVK1-CT-2001-00092-Funded by the European Commission. Muenster: Centre for Environment Research (CER), University of Muenster. ISBN 3-9808840-7-4.Google Scholar
  19. Dubois, G. (1998). Spatial interpolation comparison 97: Foreword and introduction. Journal of Geographical Information and Decision Analysis, 2(2), 1–10.Google Scholar
  20. Dubois, G., & Galmarini, S. (2005). Spatial interpolation comparison SIC2004: Introduction to the exercise and overview on the results. In G. Dubois (Ed.), EUR, 2005. Automatic mapping algorithms for routine and emergency monitoring data. EUR 21595 EN—Scientific and Technical Research Series (pp. 7–18). Luxembourg: Office for Official Publications of the European Commission. ISBN 92-894-9400-X.Google Scholar
  21. Elogne, S. N., & Hristopulos, D. T. (2008). Geostatistical applications of Spartan spatial random fields. In A. Soares, M. J. Pereira, & R. Dimitrakopoulos (Eds.), geoENV VI—Geostatistics for environmental applications in series: Quantitative geology and geostatistics, vol. 15 (pp. 477–488). Berlin: Springer.CrossRefGoogle Scholar
  22. Elogne, S., Hristopulos, D., & Varouchakis, E. (2008). An application of Spartan spatial random fields in environmental mapping: Focus on automatic mapping capabilities. Stochastic Environmental Research and Risk Assessment, 22(5), 633–646.CrossRefGoogle Scholar
  23. Fasbender, D., Peeters, L., Bogaert, P., & Dassargues, A. (2008). Bayesian data fusion applied to water table spatial mapping. Water Resources Research, 44, W12422. doi: 10.1029/2008WR006921.CrossRefGoogle Scholar
  24. Fatima, Z. (2006). Estimating soil contamination with kriging interpolation method. American Journal of Applied Sciences, 3(6), 1894–1898.CrossRefGoogle Scholar
  25. Feder, J. (1988). Fractals. New York: Plenum.Google Scholar
  26. Gambolati, G., & Volpi, G. (1979). A conceptual deterministic analysis of the kriging technique in hydrology. Water Resources Research, 15(3), 625–629.CrossRefGoogle Scholar
  27. Goovaerts, P. (1997). Geostatistics for natural resources evaluation. New York: Oxford University Press.Google Scholar
  28. Gundogdu, K., & Guney, I. (2007). Spatial analyses of groundwater levels using universal kriging. Journal of Earth System Science, 116(1), 49–55.CrossRefGoogle Scholar
  29. Hengl, T. (2007). A practical guide to geostatistical mapping of environmental variables. EUR 22904 EN-Scientific and Technical Research Series. Office for Official Publications of the European Communities: Luxembourg. 143 pp.Google Scholar
  30. Hessami, M., Anctil, F., & Viau, A. A. (2001). Delaunay implementation to improve kriging computing efficiency. Computers and Geosciences, 27(2), 237–240.CrossRefGoogle Scholar
  31. Hristopulos, D. T. (2003). Spartan Gibbs random field models for geostatistical applications. SIAM Journal on Scientific Computing, 24(6), 2125–2162.CrossRefGoogle Scholar
  32. Hristopulos, D. T., & Elogne, S. N. (2007). Analytic properties and covariance functions for a new class of generalized Gibbs random fields. IΕΕΕ Transactions on Information Theory, 53(12), 4667–4679.CrossRefGoogle Scholar
  33. Hristopulos, D. T., & Elogne, S. N. (2009). Computationally efficient spatial interpolators based on Spartan spatial random fields. IEEE Transactions on Signal Processing, 57(9), 3475–3487.CrossRefGoogle Scholar
  34. Isaaks, E. H., & Srivastava, R. M. (1989). An introduction to applied geostatisics. New York: Oxford University Press.Google Scholar
  35. Kay, M., & Dimitrakopoulos, R. (2000). Integrated interpolation methods for geophysical data: Applications to mineral exploration. Natural Resources Research, 9(1), 53–64.CrossRefGoogle Scholar
  36. Kholghi, M., & Hosseini, S. (2009). Comparison of groundwater level estimation using neuro-fuzzy and ordinary kriging. Environmental Modeling and Assessment, 14(6), 729–737.CrossRefGoogle Scholar
  37. Kitanidis, P. K. (1997). Introduction to geostatistics. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  38. Krige, D. G. (1951). A statistical approach to some basic mine valuation problems on the Witwatersrand. Journal of the Chemical, Metallurgical and Mining Society of South Africa, 52, 119–139.Google Scholar
  39. Kumar, V. (2007). Optimal contour mapping of groundwater levels using universal kriging—A case study. Hydrological Sciences, 52(5), 1038–1050.CrossRefGoogle Scholar
  40. Kumar, S., Sondhi, S. K., & Phogat, V. (2005). Network design for groundwater level monitoring in Upper Bari Doab Canal tract, Punjab, India. Irrigation and Drainage, 54(4), 431–442.CrossRefGoogle Scholar
  41. Leuangthong, O., McLennan, J. A., & Deutsch, C. V. (2004). Minimum acceptance criteria for geostatistics realizations. Natural Resources Research, 13(3), 131–141.CrossRefGoogle Scholar
  42. Ling, M., Rifai, H. S., & Newell, C. J. (2005). Optimizing groundwater long-term monitoring networks using Delaunay triangulation spatial analysis techniques. Environmetrics, 16(6), 635–657.CrossRefGoogle Scholar
  43. Mandelbrot, B. B., & Van Ness, J. W. (1968). Fractional Brownian motions, fractional noises and applications. SIAM Review, 10(4), 422–437.CrossRefGoogle Scholar
  44. Matérn, B. (1960). Spatial variation. Meddelanden fran Statens Skogsforsknings-institut, 49(5), 1–144.Google Scholar
  45. Matheron, G. (1963). Principles of geostatistics. Economic Geology, 58, 1246–1266.CrossRefGoogle Scholar
  46. Matheron, G. (1969). Le krigeage universel [Universal kriging] (Vol. 1). Fontainebleau: Cahiers du Centre de Morphologie Mathematique Fontainebleau, Ecole des Mines de Paris. 83 pp.Google Scholar
  47. Matheron, G. (1971). The theory of regionalized variables and its applications. Fontainebleau, Paris: Ecole Nationale Superieure des Mines de Paris. 211 pp.Google Scholar
  48. Mendonca, C. A., & Silva, J. B. C. (1995). Interpolation of potential-field data by equivalent layer and minimum curvature: A comparative analysis. Geophysics, 60(2), 399–407.CrossRefGoogle Scholar
  49. Mulchrone, K. F. (2003). Application of delaunay triangulation to the nearest neighbour method of strain analysis. Journal of Structural Geology, 25(5), 689–702.CrossRefGoogle Scholar
  50. Myers, D. E. (2005). Spatial interpolation comparison exercise 2004: A real problem or an academic exercise. In G. Dubois (Ed.), EUR, 2005. Automatic mapping algorithms for routine and emergency monitoring data. EUR 21595 EN—Scientific and Technical Research Series (pp. 79–88). Luxembourg: Office for official publications of the European Communities. ISBN 92-894-9400-X.Google Scholar
  51. Nikroo, L., Kompani-Zare, M., Sepaskhah, A., & Fallah Shamsi, S. (2009). Groundwater depth and elevation interpolation by kriging methods in Mohr Basin of Fars province in Iran. Environmental Monitoring and Assessment, 166(1–4), 387–407.Google Scholar
  52. Okabe, A., Boots, B., & Sugihara, K. (1992). Spatial tessellations: Concepts and applications of Voronoi diagrams. New York: Wiley.Google Scholar
  53. Olea, R. & Davis, J. C. (1999). Optimizing the High Plains aquifer water-level observation network. Open File Report 1999-15, Kansas Geological Survey.Google Scholar
  54. Pardo-Iguzquiza, E., & Chica-Olmo, M. (2008). Geostatistics with the Matern semivariogram model: A library of computer programs for inference, kriging and simulation. Computers and Geosciences, 34(9), 1073–1079.CrossRefGoogle Scholar
  55. Philip, G. M., & Watson, D. F. (1986). Automatic interpolation methods for mapping piezometric surfaces. Automatica, 22(6), 753–756.CrossRefGoogle Scholar
  56. Prakash, M. R., & Singh, V. S. (2000). Network design for groundwater monitoring—A case study. Environmental Geology, 39(6), 628–632.CrossRefGoogle Scholar
  57. Pucci, A. A. J., & Murashige, J. A. E. (1987). Applications of universal kriging to an aquifer study in New Jersey. Ground Water, 25(6), 672–678.CrossRefGoogle Scholar
  58. Rouhani, S. (1986). Comparative study of ground-water mapping techniques. Ground Water, 24(2), 207–216.CrossRefGoogle Scholar
  59. Sandwell, D. (1987). Biharmonic spline interpolation of GEOS-3 and SEASAT altimeter Dta. Geophysical Research Letters, 14(2), 139–142.CrossRefGoogle Scholar
  60. Sophocleous, M. (1983). Groundwater observation network design for the Kansas groundwater management districts, U.S.A. Journal of Hydrology, 61(4), 371–389.CrossRefGoogle Scholar
  61. Sophocleous, M., Paschetto, J. E., & Olea, R. A. (1982). Ground-water network design for northwest Kansas, using the theory of regionalized variables. Ground Water, 20(1), 48–58.CrossRefGoogle Scholar
  62. Stein, M. L. (1999). Interpolation of spatial data. Some theory for kriging. New York: Springer.CrossRefGoogle Scholar
  63. Sun, Y., Kang, S., Li, F., & Zhang, L. (2009). Comparison of interpolation methods for depth to groundwater and its temporal and spatial variations in the Minqin oasis of northwest China. Environmental Modelling and Software, 24(10), 1163–1170.CrossRefGoogle Scholar
  64. Theodossiou, N., & Latinopoulos, P. (2006). Evaluation and optimisation of groundwater observation networks using the kriging methodology. Environmental Modelling and Software, 21(7), 991–1000.CrossRefGoogle Scholar
  65. Tonkin, M. J., & Larson, S. P. (2002). Kriging water levels with a regional-linear and point-logarithmic drift. Ground Water, 40(2), 185–193.CrossRefGoogle Scholar
  66. Van den Boogaart, K. G. (2005). The comparison of one-click mapping procedures for emergency. In G. Dubois (Ed.), EUR, 2005. Automatic mapping algorithms for routine and emergency monitoring data. EUR 21595 EN—Scientific and Technical Research Series (pp. 71–78). Luxembourg: Office for official publications of the European Communities. ISBN 92-894-9400-X.Google Scholar
  67. Webster, R., & Oliver, M. (2001). Geostatistics for environmental scientists: Statistics in practice. Chichester: Wiley.Google Scholar
  68. Wessel, P. (2009). A general-purpose Green’s function-based interpolator. Computers and Geosciences, 35(6), 1247–1254.CrossRefGoogle Scholar
  69. Yang, F-g, Cao, S-y, Xing-nian, L., & Yang, K-j. (2008). Design of groundwater level monitoring network with ordinary kriging. Journal of Hydrodynamics, Ser. B, 20(3), 339–346.CrossRefGoogle Scholar
  70. Yilmaz, H. M. (2007). The effect of interpolation methods in surface definition: An experimental study. Earth Surface Processes and Landforms, 32(9), 1346–1361.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Department of Mineral Resources EngineeringTechnical University of CreteChaniaGreece

Personalised recommendations