Mathematical Geosciences

, Volume 44, Issue 8, pp 945–958 | Cite as

Change of Support in Spatial Variance-Based Sensitivity Analysis

  • Nathalie Saint-Geours
  • Christian Lavergne
  • Jean-Stéphane Bailly
  • Frédéric Grelot


Variance-based global sensitivity analysis (GSA) is used to study how the variance of the output of a model can be apportioned to different sources of uncertainty in its inputs. GSA is an essential component of model building as it helps to identify model inputs that account for most of the model output variance. However, this approach is seldom applied to spatial models because it cannot describe how uncertainty propagation interacts with another key issue in spatial modeling: the issue of model upscaling, that is, a change of spatial support of model output. In many environmental models, the end user is interested in the spatial average or the sum of the model output over a given spatial unit (for example, the average porosity of a geological block). Under a change of spatial support, the relative contribution of uncertain model inputs to the variance of aggregated model output may change. We propose a simple formalism to discuss this issue within a GSA framework by defining point and block sensitivity indices. We show that the relative contribution of an uncertain spatially distributed model input increases with its correlation length and decreases with the size of the spatial unit considered for model output aggregation. The results are briefly illustrated by a simple example.


Sobol’ indices Model upscaling Change of support Regularization theory Spatial model 


  1. Brown JD, Heuvelink GBM (2007) The Data Uncertainty Engine (DUE): a software tool for assessing and simulating uncertain environmental variables. Comput Geosci 33(2):172–190 CrossRefGoogle Scholar
  2. Cariboni J, Gatelli D, Liska R, Saltelli A (2007) The role of sensitivity analysis in ecological modelling. Ecol Model 203(1–2):167–182 CrossRefGoogle Scholar
  3. Chilès J-P, Delfiner P (1999) Geostatistics, modeling spatial uncertainty. Wiley, New York CrossRefGoogle Scholar
  4. Cressie N (1993) Statistics for spatial data, revised edn. Wiley, New York Google Scholar
  5. European Commission (2009) Impact assessment guidelines. Guideline #SEC(2009) 92 Google Scholar
  6. Heuvelink GBM (1998) Uncertainty analysis in environmental modelling under a change of spatial scale. Nutr Cycl Agroecosyst 50:255–264 CrossRefGoogle Scholar
  7. Heuvelink GBM, Brus DJ, Reinds G (2010) Accounting for spatial sampling effects in regional uncertainty propagation analysis. In: Tate NJ, Fisher PF (eds) Proceedings of Accuracy2010—The ninth international symposium on spatial accuracy assessment in natural resources and environmental sciences, pp 85–88 Google Scholar
  8. Heuvelink GBM, Burgers SLGE, Tiktak A, Van Den Berg F (2010) Uncertainty and stochastic sensitivity analysis of the GeoPearl pesticide leaching model. Geoderma 155(3–4):186–192 CrossRefGoogle Scholar
  9. Journel AG, Huijbregts CJ (1978) Mining geostatistics. The Blackburn Press, Caldwell Google Scholar
  10. Lilburne L, Tarantola S (2009) Sensitivity analysis of spatial models. Int J Geogr Inf Sci 23(2):151–168 CrossRefGoogle Scholar
  11. Marrel A, Iooss B, Jullien M, Laurent B, Volkova E (2011) Global sensitivity analysis for models with spatially dependent outputs. Environmetrics 22(3):383–397 CrossRefGoogle Scholar
  12. Pettit CL, Wilson DK (2010) Full-field sensitivity analysis through dimension reduction and probabilistic surrogate models. Probab Eng Mech 25(4):380–392 CrossRefGoogle Scholar
  13. Development Core Team R (2009) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienne. ISBN 3-900051-07-0, URL
  14. Refsgaard JC, Van Der Sluijs JP, Højberg AL, Vanrolleghem PA (2007) Uncertainty in the environmental modelling process: a framework and guidance. Environ Model Softw 22:1543–1556 CrossRefGoogle Scholar
  15. Ruffo P, Bazzana L, Consonni A, Corradi A, Saltelli A, Tarantola S (2006) Hydrocarbon exploration risk evaluation through uncertainty and sensitivity analyses techniques. Reliab Eng Syst Saf 91(10–11):1155–1162 CrossRefGoogle Scholar
  16. Saint-Geours N, Bailly J-S, Grelot F, Lavergne C (2010) Is there room to optimise the use of geostatistical simulations for sensitivity analysis of spatially distributed models. In: Tate NJ, Fisher PF (eds) Proceedings of Accuracy2010—The ninth international symposium on spatial accuracy assessment in natural resources and environmental sciences, pp 81–84 Google Scholar
  17. Saint-Geours N, Bailly J-S, Grelot F, Lavergne C (2011) Analyse de sensibilité de Sobol d’un modèle spatialisé pour l’évaluation économique du risque d’inondation. J Soc Fr Stat 152:24–46 Google Scholar
  18. Saltelli A, Ratto M, Andres T, Campolongo F, Cariboni J, Gatelli D, Saisana M, Tarantola S (eds) (2008) Global sensitivity analysis, the primer. Wiley, New York Google Scholar
  19. Schlather M (2001) Simulation of stationary and isotropic random fields. R-News 1(2):18–20 Google Scholar
  20. Sobol’ I (1993) Sensitivity analysis for non-linear mathematical model. Math Model Comput Exp 1:407–414 Google Scholar
  21. Tarantola S, Giglioli N, Jesinghaus J, Saltelli A (2002) Can global sensitivity analysis steer the implementation of models for environmental assessments and decision-making? Stoch Environ Res Risk Assess 16:63–76 CrossRefGoogle Scholar
  22. US Environmental Protection Agency (2009) Guidance on the development, evaluation, and application of environmental models. Council for Regulatory Environmental Modeling. Accessed 17 Apr 2012

Copyright information

© International Association for Mathematical Geosciences 2012

Authors and Affiliations

  • Nathalie Saint-Geours
    • 1
    • 2
  • Christian Lavergne
    • 1
  • Jean-Stéphane Bailly
    • 2
    • 3
  • Frédéric Grelot
    • 4
  1. 1.Université Montpellier 3I3MMontpellierFrance
  2. 2.AgroParisTechUMR TETISMontpellierFrance
  3. 3.AgroParisTechUMR LISAHMontpellierFrance
  4. 4.IrsteaUMR G-EAUMontpellierFrance

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