Geographically weighted methods and their use in network re-designs for environmental monitoring

  • Paul Harris
  • Annemarie Clarke
  • Steve Juggins
  • Chris Brunsdon
  • Martin Charlton
Original Paper


Given an initial spatial sampling campaign, it is often of importance to conduct a second, more targeted campaign based on the properties of the first. Here a network re-design modifies the first one by adding and/or removing sites so that maximum information is preserved. Commonly, this optimisation is constrained by limited sampling funds and a reduced sample network is sought. To this extent, we demonstrate the use of geographically weighted methods combined with a location-allocation algorithm, as a means to design a second-phase sampling campaign in univariate, bivariate and multivariate contexts. As a case study, we use a freshwater chemistry data set covering much of Great Britain. Applying the two-stage procedure enables the optimal identification of a pre-specified number of sites, providing maximum spatial and univariate/bivariate/multivariate water chemistry information for the second campaign. Network re-designs that account for the buffering capacity of a freshwater site to acidification are also conducted. To complement the use of basic methods, robust alternatives are used to reduce the effect of anomalous observations on the re-designs. Our non-stationary re-design framework is general and provides a relatively simple and a viable alternative to geostatistical re-design procedures that are commonly adopted. Particularly in the multivariate case, it represents an important methodological advance.


Non-stationarity Summary statistics PCA Location-allocation Robust Acidification 



For Harris and Charlton, research presented in this paper was funded by a Strategic Research Cluster grant (07/SRC/I1168) by the Science Foundation Ireland under the National Development Plan.


  1. Baume OP, Gebhardt A, Gebhardt C, Heuvelink GBM, Pilz J (2011) Network optimization algorithms and scenarios in the context of automatic mapping. Comput Geosci 37:289–294CrossRefGoogle Scholar
  2. Brunsdon C, Fotheringham AS, Charlton M (1998) Geographically weighted regression: modelling spatial non-stationarity. J R Stat Soc D-Sta 47:431–443CrossRefGoogle Scholar
  3. Brunsdon C, Fotheringham AS, Charlton M (2002) Geographically weighted summary statistics—a framework for localised exploratory data analysis. Comput Environ Urban 26:501–524CrossRefGoogle Scholar
  4. Brus DJ, de Gruijter J (1997) Random sampling or geostatistical modelling? Choosing between design-based and model-based sampling strategies for soil. Geoderma 80:1–59CrossRefGoogle Scholar
  5. Brus DJ, Heuvelink GBM (2007) Optimization of sample patterns for universal kriging of environmental variables. Geoderma 138:86–95CrossRefGoogle Scholar
  6. Caselton WF, Zidek JV (1984) Optimal monitoring networks. Stat Probabil Lett 2:223–227CrossRefGoogle Scholar
  7. Chilès JP, Delfiner P (1999) Geostatistics—modelling spatial uncertainty. Wiley, New YorkGoogle Scholar
  8. CLAG Freshwaters (1995) Critical loads of acid deposition for United Kingdom freshwaters. Critical Loads Advisory Group, Sub-report on Freshwaters, ITE, PenicuikGoogle Scholar
  9. Cressie NA (1989) The many faces of spatial prediction. In: Armstrong M (ed) Geostatistics, vol 1. Kluwer, Dordrecht, pp 163–176CrossRefGoogle Scholar
  10. de Gruijter J, Brus D, Bierkens M, Knotters M (2006) Sampling for natural resource monitoring. Springer, New YorkCrossRefGoogle Scholar
  11. Delmelle EM, Goovaerts P (2009) Second-phase sampling designs for non-stationary spatial variables. Geoderma 153:205–216CrossRefGoogle Scholar
  12. Diggle PJ, Menezes R, Su T (2010) Geostatistical inference under preferential sampling. J R Stat Soc C-Sta 59:191–232CrossRefGoogle Scholar
  13. Farber S, Páez A (2007) A systematic investigation of cross-validation in GWR model estimation: empirical analysis and Monte Carlo simulations. J Geogr Syst 9:371–396CrossRefGoogle Scholar
  14. Filzmoser P, Todorov V (2012) Robust tools for the imperfect world. Inform Sci. doi:  10.1016/j.ins.2012.10.017
  15. Fotheringham AS, Brunsdon C, Charlton M (2002) Geographically Weighted Regression—the analysis of spatially varying relationships. Wiley, ChichesterGoogle Scholar
  16. Gelfand AE, Sahu SK, Holland DM (2012) On the effect of preferential sampling in spatial prediction. Environmetrics 23:565–578CrossRefGoogle Scholar
  17. Glennon M, Harris P, Finne T, Scanlon R, O’Connor P (2014) Geochemical baseline for heavy metals in topsoils in Dublin, Ireland: spatial correlation with historic industry and implications for human health. Environ Geochem Health. doi:  10.1007/s10653-013-9561-8
  18. Griffith DA (2005) Effective geographic sample size in the presence of spatial autocorrelation. Ann Assoc Am Geogr 95:740–760CrossRefGoogle Scholar
  19. Haas TC (1992) Redesigning continental-scale monitoring networks. Atmos Environ 26A:3323–3333CrossRefGoogle Scholar
  20. Haas TC (2002) New systems for modelling, estimating, and predicting a multivariate spatio-temporal process. Environmetrics 13:311–332CrossRefGoogle Scholar
  21. Hampel FR (1974) The influence curve and its role in robust estimation. J Am Stat Assoc 69:383–393CrossRefGoogle Scholar
  22. Harris P, Charlton M, Fotheringham AS (2010) Moving window kriging with geographically weighted variograms. Stoch Environ Res Risk Assess 24:1193–1209CrossRefGoogle Scholar
  23. Harris P, Brunsdon C, Charlton M (2011) Geographically weighted principal components analysis. Int J Geogr Inf Sci 25:1717–1736CrossRefGoogle Scholar
  24. Harris P, Charlton M, Brunsdon C (2012) Geographically weighted (GW) models: advances in modelling spatial heterogeneity. geoENV 2012, Valencia, SpainGoogle Scholar
  25. Harris P, Brunsdon C, Charlton M, Juggins S, Clarke A (2014) Multivariate spatial outlier detection using robust geographically weighted methods. Math Geosci DOI  10.1007/s11004-013-9491-0
  26. Henriksen A, Kämäri J, Posch M, Wilander A (1992) Critical loads of acidity: Nordic surface waters. Ambio 21:356–363Google Scholar
  27. Holmes JF, Williams FB, Brown LA (1972) Faculty location under a maximum travel restriction: an example using day care facilities. Geogr Anal 4:258–266CrossRefGoogle Scholar
  28. Hornung M, Bull KR, Cresser M, Ullyett J, Hall JR, Langan S, Loveland PJ, Wilson MJ (1995) The sensitivity of surface waters of Great Britain to acidification predicted from catchment characteristics. Environ Pollut 87:207–214CrossRefGoogle Scholar
  29. Jolliffe IT (2002) Principal components analysis, 2nd edn. Springer, New YorkGoogle Scholar
  30. Journel AG (1986) Geostatistics: models and tools for the earth sciences. Math Geol 18:119–140CrossRefGoogle Scholar
  31. Kanaroglou PS, Jerrett M, Morrison J, Beckerman B, Arain MA, Gilbert NL, Brook JR (2005) Establishing an air pollution monitoring network for intra-urban population exposure assessment: a location-allocation approach. Atmos Environ 39:2399–2409CrossRefGoogle Scholar
  32. Kreiser AM, Patrick ST, Battarbee RW (1993) Critical loads for UK freshwaters—introduction, sampling strategy and use of maps. In: Hornung, M, Skeffington RA (eds) Critical loads: concepts and applications. ITE symposium No. 28, HMSO, London, pp 94–98Google Scholar
  33. Le ND, Zidek JV (1992) Interpolation with uncertain spatial covariances: a Bayesian alternative to kriging. J Multivar Anal 43:351–374CrossRefGoogle Scholar
  34. Le ND, Zidek JV (2006) Statistical analysis of environmental space–time processes. Springer, New YorkGoogle Scholar
  35. Lu B, Harris P, Gollini I, Charlton M, Brunsdon C (2013) GWmodel: an R package for exploring spatial heterogeneity. GISRUK 2013, Liverpool, UKGoogle Scholar
  36. Marchant BP, Newman S, Corstanje R, Reddy KR, Osborne TZ, Lark RM (2009) Spatial monitoring of a non-stationary soil property: phosphorus in a Florida water conservation area. Eur J Soil Sci 60:759–769CrossRefGoogle Scholar
  37. Maronna R, Martin D, Yohai V (2006) Robust statistics: theory and methods. Wiley, TorontoCrossRefGoogle Scholar
  38. Martin RJ (2001) Comparing and contrasting some environmental and experimental design problems. Environmetrics 12:273–287CrossRefGoogle Scholar
  39. McBratney AB, Webster R, Burgess TM (1981) The design of optimal sampling schemes for local estimation and mapping of regionalized variables. I. Theory and method. Comput Geosci 7:331–334CrossRefGoogle Scholar
  40. Müller WG (2005) A comparison of spatial design methods for correlated observations. Environmetrics 16:495–505CrossRefGoogle Scholar
  41. Müller WG (2007) Collecting spatial data. Springer, HeidelbergGoogle Scholar
  42. Müller WG, Zimmerman DL (1999) Optimal designs for variogram estimation. Environmetrics 10:23–37CrossRefGoogle Scholar
  43. Olea RA (2007) Declustering of clustered preferential sampling for histogram and semivariogram inference. Math Geol 39:453–467CrossRefGoogle Scholar
  44. Pretty JN, Mason CF, Nedwell DB, Hine RE, Leaf S, Dils R (2003) Environmental costs of freshwater eutrophication in England and Wales. Environ Sci Technol 37:201–208CrossRefGoogle Scholar
  45. ReVelle CS, Eiselt HA (2005) Location analysis: a synthesis and survey. Eur J Oper Res 165:1–19CrossRefGoogle Scholar
  46. Ritter K (1996) Asymptotic optimality of regular sequence designs. Ann Stat 24:2081–2096CrossRefGoogle Scholar
  47. Rosing K, Hillsman E, Rosing-Vogelaar H (1979) The robustness of two common heuristics for the P-median problem. Environ Plann A 11:373–380CrossRefGoogle Scholar
  48. Rousseeuw PJ, Debruyne M, Engelen S, Hubert M (2006) Robustness and outlier detection in chemometrics. Crit Rev Anal Chem 36:221–242CrossRefGoogle Scholar
  49. Royle JA, Nychaka D (1998) An algorithm for the construction of spatial coverage designs with implementation in SPLUS. Comput Geosci 24:479–488CrossRefGoogle Scholar
  50. Teitz MB, Bart P (1968) Heuristic methods for estimating the generalized vertex median of a weighted graph. Oper Res 16:955–961CrossRefGoogle Scholar
  51. van Groenigen J-W, Stein A (1998) Constrained optimisation of spatial sampling using continuous simulated annealing. J Environ Qual 27:1078–1086CrossRefGoogle Scholar
  52. van Groenigen J-W, Siderius W, Stein A (1999) Constrained optimisation of soil sampling for minimisation of the kriging variance. Geoderma 87:239–259CrossRefGoogle Scholar
  53. van Groenigen J-W, Pieters G, Stein A (2000) Optimizing spatial sampling for multivariate contamination in urban areas. Environmetrics 11:227–244CrossRefGoogle Scholar
  54. Varmuza K, Filzmoser P (2009) Introduction to multivariate statistical analysis in chemometrics. CRC press, New YorkCrossRefGoogle Scholar
  55. Vašát R, Heuvelink GBM, Borůvka L (2010) Sampling design optimization for multivariate soil mapping. Geoderma 155:147–153CrossRefGoogle Scholar
  56. Wang J, Haining R, Cao Z (2010) Sample surveying to estimate the mean of a heterogeneous surface: reducing the error variance through zoning. Int J Geogr Inf Sci 24:532–543Google Scholar
  57. Wang J, Stein A, Gao B, Ge Y (2012) A review of spatial sampling. Spat Stat 2:1–14CrossRefGoogle Scholar
  58. Warrick AW, Myers DE (1987) Optimisation of sampling locations for variogram calculations. Water Resour Res 23:496–500CrossRefGoogle Scholar
  59. Webster R, Welham SJ, Potts JM, Oilver MA (2006) Estimating the spatial scales of regionalized variables by nested sampling, hierarchical analysis of variance and residual maximum likelihood. Comput Geosci 32:1320–1333CrossRefGoogle Scholar
  60. Xia G, Miranda M, Gelfand AE (2006) Approximately optimal spatial design approaches for environmental health data. Environmetrics 17:363–385CrossRefGoogle Scholar
  61. Zhu Z, Stein ML (2005) Spatial sampling design for parameter estimation of the covariance function. J Stat Plann Inference 134:583–603CrossRefGoogle Scholar
  62. Zidek JV, Sun W, Le ND (2000) Designing and integrating composite networks for monitoring multivariate Gaussian pollution fields. J R Stat Soc C-Sta 49:63–79CrossRefGoogle Scholar
  63. Zimmerman DL (2006) Optimal network design for spatial prediction, covariance parameter estimation, and empirical prediction. Environmetrics 17:635–652CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Paul Harris
    • 4
  • Annemarie Clarke
    • 2
  • Steve Juggins
    • 3
  • Chris Brunsdon
    • 1
  • Martin Charlton
    • 1
  1. 1.National Centre for GeocomputationNational University of Ireland MaynoothMaynooth, Co. KildareIreland
  2. 2.APEM LtdLlantrisantUK
  3. 3.School of Geography, Politics and SociologyUniversity of NewcastleNewcastle upon TyneUK
  4. 4.Sustainable Soil and Grassland SystemsRothamsted Research, North WykeOkehampton, DevonUK

Personalised recommendations