Mathematical Geosciences

, Volume 51, Issue 4, pp 485–526 | Cite as

Geostatistics for Compositional Data: An Overview

  • Raimon Tolosana-DelgadoEmail author
  • Ute Mueller
  • K. Gerald van den Boogaart
Teaching Aid


This paper presents an overview of results for the geostatistical analysis of collocated multivariate data sets, whose variables form a composition, where the components represent the relative importance of the parts forming a whole. Such data sets occur most often in mining, hydrogeochemistry and soil science, but the results gathered here are relevant for any regionalised compositional data set. The paper covers the basic definitions, the analysis of the spatial codependence between components, mapping methods of cokriging and cosimulation honoring compositional constraints, the role of pre- and post-transformations such as log-ratios or multivariate normal score transforms, and block-support upscaling. The main result is that multivariate geostatistical techniques can and should be performed on log-ratio scores, in which case the system data-variograms-cokriging/cosimulation is intrinsically consistent, delivering the same results regardless of which log-ratio transformation was used to represent them. Proofs of all statements are included in an appendix.


Simplex Variogram Cokriging Co-simulation 



This paper was compiled during research visits between Perth and Freiberg within the project “CoDaBlockCoEstimation”, jointly funded by the German Academic Exchange Service (DAAD) and Universities Australia. The authors warmly thank Vera Pawlwowsky-Glahn and Juan José Egozcue for their constructive comments on a previous version of this manuscript.


  1. Aitchison J (1982) The statistical analysis of compositional data (with discussion). J R Stat Soc Ser B (Stat Methodol) 44:139–177Google Scholar
  2. Aitchison J (1986) The statistical analysis of compositional data. Monographs on statistics and applied probability. Chapman & Hall Ltd., London (Reprinted in 2003 with additional material by The Blackburn Press)CrossRefGoogle Scholar
  3. Aitchison J, Barceló-Vidal C, Martín-Fernández JA, Pawlowsky-Glahn V (2000) Logratio analysis and compositional distance. Math Geol 32(3):217–275CrossRefGoogle Scholar
  4. Angerer T, Hagemann S (2010) The BIF-hosted high-grade iron ore deposits in the Archean Koolyanobbing Greenstone Belt, Western Australia: structural control on synorogenic- and weathering-related magnetite-, hematite- and goethite-rich iron ore. Econ Geol 105(3):917–945CrossRefGoogle Scholar
  5. Barceló-Vidal C (2003) When a data set can be considered compositional? In: Thió-Henestrosa S, Martin-Fernández JA (eds) Proceedings of CoDaWork’03, The 1st Compositional Data Analysis Workshop. Universitat de GironaGoogle Scholar
  6. Barceló-Vidal C, Martín-Fernández JA (2016) The mathematics of compositional analysis. Austrian J Stat 45(4):57–71CrossRefGoogle Scholar
  7. Barnett RM, Manchuk JG, Deutsch CV (2014) Projection pursuit multivariate transform. Math Geosci 46(2):337–360CrossRefGoogle Scholar
  8. Bivand RS, Pebesma E, Gomez-Rubio V (2013) Applied spatial data analysis with R. Springer, New YorkCrossRefGoogle Scholar
  9. Chayes F (1960) On correlation between variables of constant sum. J Geophys Res 65(12):4185–4193CrossRefGoogle Scholar
  10. Chilès JP, Delfiner P (1999) Geostatistics. Wiley, New YorkCrossRefGoogle Scholar
  11. Cressie N (1991) Statistics for spatial data. Wiley, New YorkGoogle Scholar
  12. Egozcue JJ, Pawlowsky-Glahn V, Mateu-Figueras G, Barceló-Vidal C (2003) Isometric logratio transformations for compositional data analysis. Math Geosci 35(3):279–300Google Scholar
  13. Filzmoser P, Hron K (2008) Outlier detection for compositional data using robust methods. Math Geosci 40(3):233–248CrossRefGoogle Scholar
  14. Geovariances (2017) Isatis geostatistical software. Avon, FranceGoogle Scholar
  15. Griffin AC (1981) Structure and Iron Ore deposition in the Archaean Koolyanobbing Greenstone belt, Western Australia. Geol Soc Austra Spec Publ 7:429–438Google Scholar
  16. Journel AG, Huijbregts CJ (1978) Mining geostatistics. Academic Press, LondonGoogle Scholar
  17. Lark RM, Bishop TFA (2007) Cokriging particle size fractions of the soil. Eur J Soil Sci 58(3):763–774CrossRefGoogle Scholar
  18. Leuangthong O, Deutsch CV (2003) Stepwise conditional transformation for simulation of multiple variables. Math Geol 35(2):155–173CrossRefGoogle Scholar
  19. Mateu-Figueras G, Pawlowsky-Glahn V, Egozcue JJ (2011) The principle of working on coordinates. In: Pawlowsky-Glahn V, Buccianti A (eds) Compositional data analysis: theory and applications. Wiley, New York, pp 29–42CrossRefGoogle Scholar
  20. Mateu-Figueras G, Pawlowsky-Glahn V, Egozcue JJ (2013) The normal distribution in some constrained sample spaces. Stat Oper Res Trans 37(1):29–56Google Scholar
  21. Matheron G (1963) Principles of geostatistics. Econ Geol 58:1246–1266CrossRefGoogle Scholar
  22. Matheron G (1965) Les variables régionalisées et leur estimation-une application de la théorie des fonctions aléatoires aux sciences de la nature. Masson et Cie, ParisGoogle Scholar
  23. Matheron G (1971) The theory of regionalized variables and its applications. Technical Report C-5, École Nationale Supérieure des Mines de Paris, Centre de Geostatistique et de Morphologie Mathematique, FontainebleauGoogle Scholar
  24. Molayemat H, Torab FM, Pawlowsky-Glahn V, Hossein Morshedy A, Egozcue JJ (2018) The impact of the compositional nature of data on coal reserve evaluation, a case study in Parvadeh IV coal deposit, Central Iran. Int J Coal Geol 188:94–111. CrossRefGoogle Scholar
  25. Morales Boezio MN (2010) Estudo das metodologias alternativas da geoestatística multivariada aplicadas a estimativa de teores de depósitos de ferro. Ph. D. thesis, Universidade Federal do Rio Grande do SulGoogle Scholar
  26. Morales Boezio MN, Costa JF, Koppe JC (2012) Cokrigagem de razōes logarítmicas aditivas (alr) na estimativa de teores em depós de ferro (Cokriging of additive log-ratios (alr) for grade estimation in iron ore deposits. Rev Escol Minas 65:401–412CrossRefGoogle Scholar
  27. Mueller UA, Grunsky EC (2016) Multivariate spatial analysis of lake sediment geochemical data; Melville Peninsula, Nunavut, Canada. Appl Geochem 75:247–262. CrossRefGoogle Scholar
  28. Mueller U, Tolosana-Delgado R, van den Boogaart KG (2014) Simulation of compositional data: a nickel-laterite case study. In: Dimitrakopoulos R (ed) Advances in orebody modelling and strategic mine planning. AusIMM, MelbourneGoogle Scholar
  29. Myers DE (1982) Matrix formulation of co-kriging. Math Geol 14(3):49–257CrossRefGoogle Scholar
  30. Pawlowsky V (1984) On spurious spatial covariance between variables of constant sum. Sci Terre Sér Inform 21:107–113Google Scholar
  31. Pawlowsky V (1986) Räumliche Strukturanalyse und Schätzung ortsabhängiger Kompositionen mit Anwendungsbeispielen aus der Geologie. Ph.D. thesis, Freie Universität BerlinGoogle Scholar
  32. Pawlowsky V (1989) Cokriging of regionalized compositions. Math Geol 21(5):513–521CrossRefGoogle Scholar
  33. Pawlowsky-Glahn V (2003) Statistical modelling on coordinates. In: Thió-Henestrosa S, Martin-Fernández JA (eds) Proceedings of CoDaWork’03, The 1st Compositional Data Analysis Workshop. Universitat de GironaGoogle Scholar
  34. Pawlowsky-Glahn V, Burger H (1992) Spatial structure analysis of regionalized compositions. Math Geol 24(6):675–691CrossRefGoogle Scholar
  35. Pawlowsky-Glahn V, Egozcue JJ (2001) Geometric approach to statistical analysis on the simplex. Stoch Environ Res Risk Assess 15(5):384–398CrossRefGoogle Scholar
  36. Pawlowsky-Glahn V, Egozcue JJ (2002) BLU estimators and compositional data. Math Geol 34(3):259–274CrossRefGoogle Scholar
  37. Pawlowsky-Glahn V, Egozcue JJ (2016) Spatial analysis of compositional data: a historical review. J Geochem Explor 164:28–32. CrossRefGoogle Scholar
  38. Pawlowsky-Glahn V, Egozcue JJ, Tolosana-Delgado R (2015) Modeling and analysis of compositional data. Wiley, ChichesterGoogle Scholar
  39. Pawlowsky-Glahn V, Olea RA (2004) Geostatistical analysis of compositional data. Studies in mathematical geology 7. Oxford University Press, OxfordGoogle Scholar
  40. Pawlowsky-Glahn V, Olea RA, Davis JC (1995) Estimation of regionalized compositions: a comparison of three methods. Math Geol 27(1):105–127CrossRefGoogle Scholar
  41. Rossi ME, Deutsch CV (2014) Mineral resource estimation. Springer, New YorkCrossRefGoogle Scholar
  42. Sun XL, Wu YJ, Wang HL, Zhao YG, Zhang GL (2014) Mapping soil particle size fractions using compositional kriging, cokriging and additive log-ratio cokriging in two case studies. Math Geosci 46(4):429–443CrossRefGoogle Scholar
  43. Tjelmeland H, Lund KV (2003) Bayesian modelling of spatial compositional data. J Appl Stat 30(1):87–100CrossRefGoogle Scholar
  44. Tolosana-Delgado R (2006) Geostatistics for constrained variables: positive data, compositions and probabilities. Application to environmental hazard monitoring. Ph.D. thesis, Universitat de GironaGoogle Scholar
  45. Tolosana-Delgado R, Egozcue JJ, Pawlowsky-Glahn V (2008) Cokriging of compositions: log ratios and unbiasedness. In: Ortiz JM, Emery X (eds) Geostatistics Chile 2008. Gecamin Ltd., Santiago, pp 299–308Google Scholar
  46. Tolosana-Delgado R, Mueller U, van den Boogaart KG, Ward C (2013) Block cokriging of a whole composition. In: Costa JF, Koppe J, Peroni R (eds) Proceedings of the 36th APCOM international symposium on the applications of computers and operations research in the mineral industry. Fundacao Luiz Englert, Porto Alegre, pp 267–277Google Scholar
  47. Tolosana-Delgado R, Mueller U, van den Boogaart KG, Ward C, Gutzmer J (2015) Improving processing by adaption to conditional geostatistical simulation of block compositions. J South Afr Inst Min Metall 115(1):13–26CrossRefGoogle Scholar
  48. Tolosana-Delgado R, Otero N, Pawlowsky-Glahn V (2005) Some basic concepts of compositional geometry. Math Geol 37(7):673–680CrossRefGoogle Scholar
  49. Tolosana-Delgado R, van den Boogaart KG (2013) Joint consistent mapping of high-dimensional geochemical surveys. Math Geosci 45(8):983–1004CrossRefGoogle Scholar
  50. Tolosana-Delgado R, van den Boogaart KG, Pawlowsky-Glahn V (2011) Geostatistics for compositions. In: Pawlowsky-Glahn V, Buccianti A (eds) Compositional data analysis: theory and applications. Wiley, New York, pp 73–86CrossRefGoogle Scholar
  51. van den Boogaart KG, Tolosana-Delgado R, Bren M (2018) Compositions: compositional data analysis package R package version 1.40-2Google Scholar
  52. van den Boogaart KG, Tolosana-Delgado R (2013) Analysing compositional data with R. Springer, HeidelbergCrossRefGoogle Scholar
  53. van den Boogaart KG, Tolosana-Delgado R, Mueller U (2017) An affine equivariant multivariate normal score transform for compositional data. Math Geosci 49(2):231–252CrossRefGoogle Scholar
  54. Wackernagel H (2003) Multivariate geostatistics: an introduction with applications. Springer, BerlinCrossRefGoogle Scholar
  55. Walwoort DJ, de Gruijter JJ (2001) Compositional kriging: a spatial interpolation method for compositional data. Math Geol 33(8):951–966CrossRefGoogle Scholar
  56. Ward C, Mueller U (2012) Multivariate estimation using log ratios: a worked alternative. In: Abrahamsen P, Hauge R, Kolbjornsen O (eds) Geostatistics Oslo 2012. Springer, Berlin, pp 333–343CrossRefGoogle Scholar
  57. Ward C, Mueller U (2013) Compositions, log ratios and bias—from grade control to resource. In: Iron Ore 2013 shifting the paradigm. AUSIMM, Melbourne, pp 313–320Google Scholar

Copyright information

© International Association for Mathematical Geosciences 2018

Authors and Affiliations

  1. 1.Helmholtz-Zentrum Dresden-Rossendorf, Helmholtz Institute Freiberg for Resource TechnologyFreibergGermany
  2. 2.School of ScienceEdith Cowan UniversityJoondalupAustralia
  3. 3.Institut für StochastikTechnische Universität “Bergakademie” FreibergFreibergGermany

Personalised recommendations