A new approach to spatial data interpolation using higher-order statistics

  • Shen Liu
  • Vo Anh
  • James McGree
  • Erhan KozanEmail author
  • Rodney C. Wolff
Original Paper


Interpolation techniques for spatial data have been applied frequently in various fields of geosciences. Although most conventional interpolation methods assume that it is sufficient to use first- and second-order statistics to characterize random fields, researchers have now realized that these methods cannot always provide reliable interpolation results, since geological and environmental phenomena tend to be very complex, presenting non-Gaussian distribution and/or non-linear inter-variable relationship. This paper proposes a new approach to the interpolation of spatial data, which can be applied with great flexibility. Suitable cross-variable higher-order spatial statistics are developed to measure the spatial relationship between the random variable at an unsampled location and those in its neighbourhood. Given the computed cross-variable higher-order spatial statistics, the conditional probability density function is approximated via polynomial expansions, which is then utilized to determine the interpolated value at the unsampled location as an expectation. In addition, the uncertainty associated with the interpolation is quantified by constructing prediction intervals of interpolated values. The proposed method is applied to a mineral deposit dataset, and the results demonstrate that it outperforms kriging methods in uncertainty quantification. The introduction of the cross-variable higher-order spatial statistics noticeably improves the quality of the interpolation since it enriches the information that can be extracted from the observed data, and this benefit is substantial when working with data that are sparse or have non-trivial dependence structures.


Geostatistics Interpolation Uncertainty quantification Mineral deposit 



The authors would like to acknowledge the support of CRC-ORE, established and supported by the Australian Government’s Cooperative Research Centres Programme. The authors are also grateful to the two anonymous referees for their valuable comments and suggestions that have contributed to improving the quality and presentation of this paper.


  1. Arpat B, Caers J (2007) Conditional simulation with patterns. Math Geosci 39:177–203Google Scholar
  2. Babak O (2013) Inverse distance interpolation for facies modeling. Stoch Environ Res Risk Assess. doi: 10.1007/s00477-013-0833-8 Google Scholar
  3. Babak O, Deutsch CV (2009) Statistical approach to inverse distance interpolation. Stoch Environ Res Risk Assess 23:543–553CrossRefGoogle Scholar
  4. Bárdossy A (2006) Copula-based geostatistical models for groundwater quality parameters. Water Resour Res 42:W11416Google Scholar
  5. Bárdossy A, Li J (2008) Geostatistical interpolation using copulas. Water Resour Res 44:W07412CrossRefGoogle Scholar
  6. Boucher A (2009) Considering complex training images with search tree partitioning. Comput Geosci 35:1151–1158CrossRefGoogle Scholar
  7. Chatterjee S, Dimitrakopoulos R (2012) Multi-scale stochastic simulation with a wavelet-based approach. Comput Geosci 45:177–189CrossRefGoogle Scholar
  8. Chilès JP, Delfiner P (1999) Geostatistics—modeling spatial uncertainty. Wiley, New YorkGoogle Scholar
  9. Chugunova TL, Hu LY (2008) Multiple point simulations constrained by continuous auxiliary data. Math Geosci 40:133–146CrossRefGoogle Scholar
  10. Cressie N (1985) Fitting variogram models by weighted least squares. Math Geol 17:563–586CrossRefGoogle Scholar
  11. De Iaco S (2013) On the use of different metrics for assessing complex pattern reproductions. J Appl Stat. doi: 10.1080/02664763.2012.754853 Google Scholar
  12. Dimitrakopoulos R, Mustapha H, Gloaguen E (2010) High-order statistics of spatial random fields: exploring spatial cumulants for modelling complex, non-Gaussian and non-linear phenomena. Math Geosci 42:65–99CrossRefGoogle Scholar
  13. Gaetan C, Guyon X (2010) Spatial statistics and modeling. Springer, New YorkCrossRefGoogle Scholar
  14. Gloaguen E, Dimitrakopoulos R (2008) Conditional wavelet based simulation of nonstationary geology using geophysical and model analogue information. Paper presented at the Proceedings of Geostats 2008—VIII International Geostatistics Congress, Santiago, ChileGoogle Scholar
  15. Gloaguen E, Dimitrakopoulos R (2009) Two-dimensional conditional simulations based on the wavelet decomposition of training images. Math Geosci 41:679–701CrossRefGoogle Scholar
  16. Goodfellow R, Mustapha H, Dimitrakopoulos R (2012) Approximations of high-order spatial statistics through decomposition. Quant Geol Geostat, Geostat Oslo 17:91–102. doi: 10.1007/978-94-007-4153-9_8 CrossRefGoogle Scholar
  17. Honarkhah M, Caers J (2010) Stochastic simulation of patterns using distance-based pattern modelling. Math Geosci 42:487–517CrossRefGoogle Scholar
  18. Hosny KM (2007) Exact Legendremoment computation for gray level images. Pattern Recogn 40:3597–3605CrossRefGoogle Scholar
  19. Hwang Y, Clark M, Rajagopalan B, Leavesley G (2012) Spatial interpolation schemes of daily precipitation for hydrologic modeling. Stoch Environ Res Risk Assess 26:295–320CrossRefGoogle Scholar
  20. Kazianka H (2013) Approximate copula-based estimation and prediction of discrete spatial data. Stoch Environ Res Risk Assess 27:2015–2026CrossRefGoogle Scholar
  21. Kazianka H, Pilz J (2010a) Copula-based geostatistical modeling of continuous and discrete data including covariates. Stoch Environ Res Risk Assess 24:661–673CrossRefGoogle Scholar
  22. Kazianka H, Pilz J (2010b) Spatial interpolation using copula-based geostatistical models. In: Atkinson P, Lloyd C (eds) geoENV VII—Geostatistics for environmental applications. Springer, BerlinGoogle Scholar
  23. Kazianka H, Pilz J (2011) Bayesian spatial modeling and interpolation using copulas. Comput Geosci 37:310–319CrossRefGoogle Scholar
  24. Kazianka H, Pilz J (2012) Objective Bayesian analysis of spatial data with uncertain nugget and range parameters. Can J Stat 40:304–327CrossRefGoogle Scholar
  25. Kim JH, Wong K, Athanasopoulos G, Liu S (2011) Beyond point forecasting: evaluation of alternative prediction intervals for tourist arrivals. Int J Forecasting 27:887–901CrossRefGoogle Scholar
  26. Kleijnen JPC, Mehdad E, Beers WCM (2012) Convex and monotonic bootstrapped kriging. Proceedings of the 2012 Winter Simulation ConferenceGoogle Scholar
  27. Li M, Shao Q, Renzullo L (2010) Estimation and spatial interpolaiton of rainfall intensity distribution form the effective rate of precipitation. Stoch Environ Res Risk Assess 24:117–130CrossRefGoogle Scholar
  28. Liu J, Spiegel MR (1999) Mathematical handbook of formulas and tables, 2nd edn. McGraw-Hill, New YorkGoogle Scholar
  29. Loh JM, Stein ML (2004) Bootstrapping a spatial point process. Stat Sinica 14:69–101Google Scholar
  30. Loh JM, Stein ML (2008) A valid and fast spatial bootstrap for correlation functions. Astrophys J 681:726–734CrossRefGoogle Scholar
  31. Machuca-Mory DF, Dimitrakopoulos R (2012) Simulation of a structurally-controlled gold deposit using high-order statistics. Paper presented at the Ninth International Geostatistics Congress, Oslo, NorwayGoogle Scholar
  32. Mirowski PW, Trtzlaff DM, Davies RC, McCormick DS, Williams N, Signer C (2008) Stationary scores on training images for multipoint geostatistics. Math Geosci 41:447–474CrossRefGoogle Scholar
  33. Mukul M, Roy D, Satpathy S, Kumar VA (2004) Bootstrapped spatial statistics: a more robust approach to the analysis of finite strain data. J Struct Geol 26:595–600CrossRefGoogle Scholar
  34. Mustapha H, Dimitrakopoulos R (2010a) A new approach for geological pattern recognition using high-order spatial cumulants. Comput Geosci 36:313–343CrossRefGoogle Scholar
  35. Mustapha H, Dimitrakopoulos R (2010b) High-order stochastic simulation of complex spatially distributed natural phenomena. Math Geosci 42:457–485CrossRefGoogle Scholar
  36. Mustapha H, Dimitrakopoulos R (2011) HOSIM: a high-order stochastic simulation algorithm for generating three-dimensional complex geological patterns. Comput Geosci 37:1242–1253CrossRefGoogle Scholar
  37. Mustapha H, Dimitrakopoulos R, Chatterjee S (2011) Geologic heterogeneity representation using high-order spatial cumulants for subsurface flow and transport simulations. Water Resour Res 47. doi:  10.1029/2010WR009515
  38. Nieto-Barajas LE, Sinha T (2014) Bayesian interpolation of unequally spaced time series. Stoch Environ Res Risk Assess. doi: 10.1007/s00477-014-0894-3 Google Scholar
  39. Pilz J, Kazianka H, Spöck G (2012) Some advances in Bayesian spatial prediction and sampling design. Spatial Stat 1:65–81CrossRefGoogle Scholar
  40. Remy N, Boucher A, Wu J (2009) Applied geostatistics with SGeMs: a users’s guide. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  41. Rojas-Avellaneda D, Silvan-Cardenas JL (2006) Performance of geostatistical interpolation methods for modeling sampled data with non-stationary mean. Stoch Environ Res Risk Assess 20:455–467CrossRefGoogle Scholar
  42. Saito H, McKenna SA, Zimmerman DA, Coburn TC (2005) Geostatistical interpolation of object counts collected from multiple strip transects: ordinary kriging versus finite domain kriging. Stoch Environ Res Risk Assess 19:71–85CrossRefGoogle Scholar
  43. Scheidt C, Caers J (2009a) Representing spatial uncertainty using distances and kernels. Math Geosci 41:397–419CrossRefGoogle Scholar
  44. Scheidt C, Caers J (2009b) Uncertainty quantification in reservoir performance using distances and kernel methods—application to a West Africa deepwater turbidite reservoir. SPE J 14:680–692CrossRefGoogle Scholar
  45. Scheidt C, Caers J (2010) Bootstrap confidence intervals for reservoir model selection techniques. Comput Geosci 14:369–382CrossRefGoogle Scholar
  46. Schelin L, Luna SS (2010) Kriging prediction intervals based on semiparametric bootstrap. Math Geosci 42:985–1000CrossRefGoogle Scholar
  47. Strebelle S (2002) Conditional simulation of complex geological structures using multiple point statistics. Math Geosci 34:1–22Google Scholar
  48. Tjelmeland H, Besag J (1998) Markov random fields with higher order interactions. Scand J Stat 25:415–433CrossRefGoogle Scholar
  49. Troldborg M, Nowak W, Lange IV, Santos MC, Binning PJ, Bjerg PL (2012) Application of Bayesian geostatistics for evaluation of mass discharge uncertainty at contaminated sites. Water Resour Res 48:W09535CrossRefGoogle Scholar
  50. Wu J, Boucher A, Zhang T (2008) SGeMS code for pattern simulation of continuous and categorical variables: FILTERSIM. Comput Geosci 34:1863–1876CrossRefGoogle Scholar
  51. Zhang T, Switzer P, Journel AG (2006) Filter-based classification of training image patterns for spatial simulation. Math Geosci 38:63–80Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Shen Liu
    • 1
    • 3
  • Vo Anh
    • 1
    • 3
  • James McGree
    • 1
    • 3
  • Erhan Kozan
    • 1
    • 3
    • 4
    Email author
  • Rodney C. Wolff
    • 2
    • 3
  1. 1.Mathematical Sciences SchoolQueensland University of TechnologyBrisbaneAustralia
  2. 2.WH Bryan Mining and Geology Research CentreThe University of QueenslandBrisbaneAustralia
  3. 3.Cooperative Research Centre for Optimised Resource Extraction (CRC-ORE)BrisbaneAustralia
  4. 4.Queensland University of TechnologyBrisbaneAustralia

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