Mathematical Geosciences

, Volume 45, Issue 4, pp 453–470 | Cite as

Automatic Variogram Modeling by Iterative Least Squares: Univariate and Multivariate Cases

  • N. Desassis
  • D. Renard


In this paper, we propose a new methodology to automatically find a model that fits on an experimental variogram. Starting with a linear combination of some basic authorized structures (for instance, spherical and exponential), a numerical algorithm is used to compute the parameters, which minimize a distance between the model and the experimental variogram. The initial values are automatically chosen and the algorithm is iterative. After this first step, parameters with a negligible influence are discarded from the model and the more parsimonious model is estimated by using the numerical algorithm again. This process is iterated until no more parameters can be discarded. A procedure based on a profiled cost function is also developed in order to use the numerical algorithm for multivariate data sets (possibly with a lot of variables) modeled in the scope of a linear model of coregionalization. The efficiency of the method is illustrated on several examples (including variogram maps) and on two multivariate cases.


Automatic fitting Variogram maps Linear model of coregionalization Anisotropy Weighted least squares Over-fitting 



This work was partially supported by French ANR CRISCO2. The authors would like to thank J.P. Chilès, N. Jeannee of Geovariances™, and one anonymous reviewer for helpful comments.


  1. Armstrong M, Galli A, Beucher H, Le Loc’h G, Renard D, Doligez B, Eschard R, Geffroy F (2011) Plurigaussian simulation in geosciences. Springer, Berlin CrossRefGoogle Scholar
  2. Chilès J, Delfiner P (2012) Geostatistics: Modeling spatial uncertainty, 2nd edn. Wiley, New York CrossRefGoogle Scholar
  3. Cressie N (1985) Fitting variogram models by weighted least squares. Math Geol 17(5):563–586 CrossRefGoogle Scholar
  4. Cressie N (1993) Statistics for spatial data. Wiley, New York Google Scholar
  5. Cressie N, Lahiri SN (1996) Asymptotics for REML estimation of spatial covariance parameters. J Stat Plan Inference 50:327–341 CrossRefGoogle Scholar
  6. Diggle PJ, Ribeiro PJ (2007) Model-based geostatistics. Springer series in statistics. Springer, New York Google Scholar
  7. Emery X (2010) Iterative algorithms for fitting a linear model of coregionalization. Comput Geosci 36:1150–1160 CrossRefGoogle Scholar
  8. de Fouquet C, Malherbe L, Ung A (2011) Geostatistical analysis of the temporal variability of ozone concentrations. Comparison between CHIMERE model and surface observations. Atmos Environ 45:3434–3446 CrossRefGoogle Scholar
  9. Goulard M, Voltz M (1992) Linear coregionalization model: Tools for estimation and choice of cross-variogram matrix. Math Geol 24:269–286 CrossRefGoogle Scholar
  10. Handcock MS, Wallis JR (1994) An approach to statistical spatial-temporal modeling of meteorological fields. J Am Stat Assoc 89(426):368–390 CrossRefGoogle Scholar
  11. Horn RA, Johnson CR (1985) Matrix analysis. Cambridge University Press, Cambridge CrossRefGoogle Scholar
  12. Isatis® (2012) Geostatistical Software by Geovariances™ Google Scholar
  13. Larrondo PF, Neufeld CT, Deutsch CV (2003) VARFIT: A program for semi-automatic variogram modeling. In: Deutsch CV (ed) Fifth annual report of the centre for computational geostatistics. University of Alberta, Edmonton, 17 pp Google Scholar
  14. Madsen K, Nielsen HB, Tingleff O (2004a) Methods for non-linear least squares problems, 2nd edn. Tech. rep., Informatics and Mathematical Modeling, Technical University, Denmark Google Scholar
  15. Madsen K, Nielsen HB, Tingleff O (2004b) Optimization with constraints, 2nd edn. Tech. rep., Informatics and Mathematical Modeling, Technical University, Denmark Google Scholar
  16. Magneron C, Jeannee N, Le Moine O, Bourillet JF (2009) Integrating prior knowledge and locally varying parameters with moving-geostatistics: Methodology and application to bathymetric mapping. In: Atkinson PM, Lloyds CD (eds) GeoEnv VII—Geostatistics for environmental applications. Springer, New York Google Scholar
  17. Mardia KV, Marshall RJ (1984) Maximum likelihood estimation of models for residual covariance in spatial regression. Biometrika 71(1):135–146 CrossRefGoogle Scholar
  18. Oman SD, Vakulenko-Lagun B (2009) Estimation of sill matrices in the linear model of coregionalization. Math Geosci 41:15–27 CrossRefGoogle Scholar
  19. Pardo-Igúzquiza E (1999) VARFIT: A FORTRAN-77 program for fitting variogram models by weighted least squares. Comput Geosci 25:251–261 CrossRefGoogle Scholar
  20. Petitgas P, Doray M, Mass J, Grellier P (2011) Spatially explicit estimation of fish length histograms with application to anchovy habitats in the bay of Biscay. ICES J Mar Sci 68(10):2086–2095 CrossRefGoogle Scholar
  21. Renard D, Bez N, Desassis N, Beucher H, Ors F (2012) RGeoS: Geostatistical package [9.0.2]. MINES-ParisTech. Free download from
  22. Stein M (1999) Interpolation of spatial data, some theory for kriging. Springer Series in Statistics. Springer, New York CrossRefGoogle Scholar
  23. Wackernagel H (2003) Multivariate geostatistics—An introduction with application, 3rd edn. Springer, New York CrossRefGoogle Scholar

Copyright information

© International Association for Mathematical Geosciences 2012

Authors and Affiliations

  1. 1.Equipe de Géostatistique, Centre de GéosciencesMINES-ParisTechFontainebleauFrance

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