A Review of Geo-Spatial Sampling Theory

  • 1. Institute of Geographic Sciences and Natural Resources Research,CAS,Beijing 100101,China;
    2. Graduate University of Chinese Academy of Sciences,Beijing 100039,China;
    3. Institute of Automation,CAS,Beijing 100080,China

Received date: 2008-10-28

  Revised date: 2008-12-29

  Online published: 2009-03-25

Supported by

National Natural Science Foundation of China,No.40471111;No70571076;863 Program,No.2006AA12Z205


Sample survey is the groundwork of the studies of natural resources, environmental problems and socio-economy. The geo-related characteristic of spatial data limits the application of classic sampling theory, which is essentially based on independent assumption. Spatial sampling theory is the foundation of sample survey of spatial related resources. Firstly, this paper introduces the history of spatial sampling theory and presents four main issues addressed by this theory. Then it reviews the theory and the applications of model-based and design-based statistics inference. Finally, this paper gives a detailed description of (1): Kriging theory application in spatial sampling; (2): forward and backward samples distribution methods, and the combination of the above two; (3): six criteria for optimization of sample selection.

Cite this article

JIANG Chengsheng,WANG Jinfeng,CAO Zhidong . A Review of Geo-Spatial Sampling Theory[J]. Acta Geographica Sinica, 2009 , 64(3) : 368 -380 . DOI: 10.11821/xb200903012


[1] Matheron G. Principles of geostatistics. Economic Geology, 1963, 58: 1246-1266.

[2] Matheron G. Kriging, or polynomial interpolation procedures. Canadian Mining and Metallurgical Bulletin, 1967, 60: 1041-1045.

[3] Tobler W. A computer movie simulating urban growth in the Detroit Region. Economic Geography, 1970, 46 (2): 234-240.

[4] Cliff A, Ord J. Spatial Autocorrelation. London: Pion, 1973.

[5] Fisher M, Scholten H J, Unwin D. Spatial Analytical Perspectives on GIS. London: Taylor & Francis, 1996.

[6] Anselin L. Spatial Econometrics: Methods and Models. Dordrecht: Kluwer Academic, 1988.

[7] Haining R P. Spatial Data Analysis: Theory and Practice. Cambridge: Cambridge University, 2003.

[8] Ripley B D. Spatial Statistics. New York: John Wiley & Sons. Inc., 1981.

[9] Griffith D A. Spatial Autocorrelation and Spatial Filtering. Springer, 2003.

[10] Stark K E, Arsenault A, Bradfield G E. Variation in soil seed bank species composition of a dry coniferous forest: Spatial scale and sampling considerations. Plant Ecology, 2008, 197(2): 173-181.

[11] Nakamoto S, Fang Z, Matsuura T. Spatial sampling requirements for tropical Pacific sea surface temperature variability. Journal of Geophysical Research, 1994, 99(C9): 18363-18370.

[12] Nick G, Howard C J, Mccormick Mark I. Spatial variability in reef fish distribution, abundance, size and biomass: A multi-scale analysis. Marine Ecology, 2001, 214: 237-251.

[13] Dessard H, Bar-Hen A. Experimental design for spatial sampling applied to the study of tropical forest regeneration. Canadian Journal of Forest Research, 2005, 35(5): 1149-1155.

[14] Lianfa L, Wang Jinfeng. Optimal decision-making model of spatial sampling for survey of China's land with remotely sensed data. Science in China (Series D), 2005, 48(6): 752-764.

[15] Kumar N. Spatial sampling design for a demographic and health survey. Population Research and Policy Review, 2007, 26(5): 581-599.

[16] Fuentes M, Chaudhuri A, Holland D. Bayesian entropy for spatial sampling design of environmental data. Environmental and Ecological Statistics, 2007, 14(3): 323-340.

[17] Lark R M. Optimized spatial sampling of soil for estimation of the variogram by maximum likelihood. Geoderma, 2002, 105(1/2): 49-80.

[18] Christakos G, Killam B R. Sampling design for classifying contaminant level using annealing search algorithms. Water Resources Research, 1993, 29(12): 4063-4076.

[19] Feng Shiyong. Some hotspot issues on the application and theory of survey sampling. Statistics & Information Forum, 2007, 22(1): 5-31.
[冯士雍. 抽样调查应用与理论中的若干前沿问题. 统计与信息论坛, 2007, 22(1): 5-31.]

[20] Feng Shiyong, Shi Xiquan. Sampling Surveysalysis: Theory, Methods and Case Analysis. Shanghai: Shanghai Scientific & Technical Publishers, 1996.
[冯士雍, 施锡铨. 抽样调查: 理论、方法与实践. 上海: 上海科学技术出版社, 1996.]

[21] Zhao Xianwen. Quantitative Methods by Remote Sensing in Forestry. Beijing: China Forestry Publishing House, 1997.
[赵宪文. 林业遥感定量估测. 北京: 中国林业出版社, 1997.]

[22] Wu Bingfang. Operational remote sensing methods for agricultural statistics. Acta Geographica Sinica, 2000, 55 (1): 25-35.
[吴炳方. 全国农情监测与估产运行化遥感方法. 地理学报, 2000, 55(1): 25-35.]

[23] Li Lianfa, Wang Jinfeng, Liu Jiyuan. Optimal decision-making model of spatial sampling for survey of China's land with remotely sensed data. Science in China (Series D), 2005, 48(5).

[24] Li Lianfa, Wang Jinfeng. Integrated spatial sampling modeling of geospatial data. Science in China (Series D), 2004, 47 (3): 201-208.

[25] Wang Jinfeng, Zhuang Dafang, Li Lianfa. Spatial sampling design for monitoring the area of cultivated land. International Journal of Remote Sensing, 2002, 13(2): 263-284.

[26] Wang Jinfeng, Robert H, Wise S. Spatial sampling design for monitoring drought, flood and earthquake in China. Progress in Natural Science, 1999, (4).
[王劲峰, Robert H, Wise S. 中国干旱洪水地震灾害监测空间采样设计. 自然 科学进展, 1999, (4).]

[27] Wang X J, Qi F. The effects of sampling design on spatial structure analysis of contaminated soil. Science of the Total Environment, 1998, 223(1-3): 29-41.

[28] Delmelle E. Optimization of second-phase spatial sampling using auxiliary information. In: Department of Geography, State University of New York at Buffalo, 2005. 108.

[29] Cooper C. Sampling and variance estimation on continuous domains. Environmetrics, 2006, 17(6): 539-553.

[30] Dalenius T, Hájek J, Zubrzycki S. On plane sampling and related geometrical problems. In: Proceedings of the 4th Berkeley Symposium on Probability and Mathematical Statistics, 1961. 125-150.

[31] Olea R A. Sampling design optimization for spatial functions. Mathematical Geology, 1984, 16: 369-392.

[32] Overton W, Stehman S. Properties of designs for sampling continuous spatial resources from a triangular grid. Communications in Statistics Part A: Theory and Methods, 1993, 22: 2641-2660.

[33] Stevens Jr D L, Olsen A R. Spatially Restricted surveys over time for aquatic resources. Journal of Agricultural, Biological, and Environmental Statistics, 1999, 4: 415-428.

[34] Stevens Jr D L, Olsen A R. Spatially-restricted random sampling designs for design-based and model-based estimation. In: Accuracy 2000: Proceedings of the 4th International Symposium on Spatial Accuracy Assessment in Natural Resources and Environmental Sciences. The Netherlands: Delft University Press, 2000. 609-616.

[35] Stevens Jr D L, Olsen A R. Spatially-balanced sampling of natural resources. Journal of the American Statistical Association, 2004, 99: 262-277.

[36] Banjevic M. Optimal network designs in spatial statistics. In: Department of Statistics, Stanford University, 2004. 114.

[37] Nunes L M. Optimal estuarine sediment monitoring network design with simulated annealing. Journal of Environmental Management, 2006, 78: 294-304.

[38] Wagner B J. Sampling design methods for groundwater modeling under uncertainty. Water Resources Research, 1995, 31(10): 2581-2591.

[39] Yan S Q, Minsker B. Optimal groundwater remediation design using an Adaptive Neural Network Genetic Algorithm. Water Resources Research, 2006. 42.

[40] Zhu Z Y. Optimal sampling design and parameter estimation of Gaussian random fields. In: Department of Statistics, The University of Chicago, 2002. 132.

[41] Zimmerman D L. Optimal network design for spatial prediction, covariance parameter estimation, and empirical prediction. Environmetrics, 2006, 17: 635-652.

[42] Brus D J, de Gruijter J J. Random sampling or geostatistical modelling? Choosing between design-based and model-based sampling strategies for soil (with discussion). Geoderma, 1997, 80: 1-44.

[43] Haining R P. Spatial Data Analysis: Theory and Practice. Cambridge: Cambridge University, 2003.

[44] S覿rndal C E, Swensson B, Wretman J. Model Assisted Survey Sampling. New York: Springer, 1992.

[45] Defeo O, Rueda M. Spatial structure, sampling design and abundance estimates in sandy beach macroinfauna: Some warnings and new perspectives. Marine Biology, 2002, 140(6): 1215-1225.

[46] Flores L A, Martinez L I, Ferrer C M. Systematic sample design for the estimation of spatial means. Environmetrics, 2003, 14(1): 45-61.

[47] Lark R M, Cullis B R. Model-based analysis using REML for inference from systematically sampled data on soil. European Journal of Soil Science, 2004, 55(4): 799-813.

[48] Little R J. To model or not to model? Competing modes of inference for finite population sampling. Journal of the American Statistical Association, 2004, 99: 546-556.

[49] Papritz A, Webster R. Estimating temporal change in soil monitoring: Sampling from simulated fields. EuropeanJournal of Soil Science, 1995, 46(1): 13-27.

[50] Cordy C B, Thompson C M. An application of the deterministic variogram to design-based variance estimation. Mathematical Geology, 1995, 27: 173-205.

[51] Jiang Baofa, Xu Xiaofei, Wang Jichuan. Time/space location sampling and its application in survey of outpatients visiting sexually transmitted diseases clinic. Chinese Journal of Health Statistics, 2003, 20(4): 502-702.
[姜宝法, 徐晓 菲, 王济川. 时间/ 空间定位抽样设计及其在性病门诊病人调查中的应用. 中国卫生统计, 2003, 20(4): 502-702.]

[52] Jin Yongjin, Hou Zhiqiang. Self-weighting sample design for the China's population changes survey. Statistics & Information Forum, 2007, 22(4): 11-13.
[金勇进, 侯志强. 中国人口变动调查的自加权抽样设计. 统计与信息论坛, 2007, 22(4): 11-13.]

[53] Cochran W G. Sampling techniques. 3rd edn. New York: John Wiley & Sons, 1977.

[54] Sukhatme B V. Testing the hypothesis that two populations differ only in location. The Annals of Mathematical Statistics, 1958, 29(1): 60-78.

[55] De Gruijter J J, Braak C J F T. Model free estimation from spatial samples: A reappraisal of classical sampling theory Mathematical Geology, 1990, 22(4): 407-415.

[56] Stevens Jr. D L. Spatial properties of design-based versus model-based approaches to environmental sampling. In: Caetano M, Painho M (eds.). 7th International Symposium on Spatial Accuracy Assessment in Natural Resources and Environmental Sciences, 2006.

[57] Cressie N A C. Statistics for Spatial Data. New York: Wiley-Interscience, 1993.

[58] Li Lianfa, Wang Jinfeng, Liu Jiyuan. Optimization of decision-making for spatial sampling in remote sensing survey of land. Science in China (Series D), 2004, (10): 975-982.
[李连发, 王劲峰, 刘纪远. 国土遥感调查的空间抽样优化决 策. 中国科学(D 辑), 2004, (10): 975-982.]

[59] Li Lianfa, Wang Jinfeng. Spatial sampling model for geographic data. Progress in Natural Science, 2002, (5).
[李连发, 王劲峰. 地理数据空间抽样模型. 自然科学进展, 2002, (5).]

[60] Niu Wenjie. Prior information residual Kriging. Computer Engineering and Applications, 2004, (35).
[牛文杰. 基于先 验信息的残余克里金法的研究. 计算机工程与应用, 2004, (35).]

[61] Zhao Xuehui. Sample survey methods and theory progress. Statistics & Information Forum , 2003, 18(5): 42-72.
[赵雪 慧. 抽样调查理论和方法的最新进展. 统计与信息论坛, 2003, 18(5): 42-72.]

[62] Li Chonggui, Zhao Xianwen. Determining the optimum sample plots for establishing canopy density estimating equation in monitoring area. Journal of Beijing Forestry University, 2005. 27(6): 24-27.
[李崇贵, 赵宪文. 确定监测区 域建立森林郁闭度估测方程最优样地的研究. 北京林业大学学报, 2005, 27(6): 24-27.]

[63] Matheron G. Les Variables Régionalisées et Leur Estimation. Paris: Masson, 1965.

[64] Matheron G. The Theory of Regionalized Variables and Its Applications. Fontainbleau: Ecole de Mines, 1971.

[65] Oliver M A, Webster R. Kriging: A method of interpolation for geographical information systems. International Journal of Geographical Information Science, 1990, 4(3): 313-332.

[66] Laslett G M, McBratney A B, Pahl P J. Comparison of several spatial prediction methods for soil pH. European Journal of Soil Science, 1987, 38: 325-341.

[67] Cressie N, Hawkins D M. Robust estimation of the variogram I. Mathematical Geology, 1980, 12(2): 115-125.

[68] Webster R, Oliver M A. How large a sample is needed to estimate the regional variogram adequately? In: Soares A (ed). Geostatistics Tro'ia '92. Dordrecht: Kluwer Academic, 1993, 155-165.

[69] Hughes J P, Lettenmaier D P. Data requirements for Kriging: Estimation and network design. Water Resources Research, 1981, 17: 1641-1650.

[70] Russo D. Design of an optimal sampling network for estimating the variogram. Soil Science Society of America Journal, 1984, 48: 708-716.

[71] Warrick A W, Myers D E. Optimization of sampling locations for variogram. Calculations Water Resources Research, 1987, 23: 496-500.

[72] Hammond L C, Pritchett W L, Chew U. Soil sampling in relation to soil heterogeneity. Soil Science Society of America Proceedings, 1958, 22: 548-552.

[73] Olson J S, Potter P E. Variance components of crossbedding direction in some basal Pennsylvanian sandstones of the Eastern Interior Basin: statistical methods. Journal of Geology, 1954, 62: 26-49.

[74] Webster R, Butler B E. Soil survey and classification studies at Ginninderra. Australian Journal of Soil Research, 1976, 14: 1-24.

[75] Yfantis E A, Flatman G T, Behar J V. Efficiency of Kriging estimation for square, triangular, and hexagonal grids. Mathematical Geology, 1987, 19(3): 183-205.

[76] Webster R et al. Estimating the spatial scales of regionalized variables by nested sampling, hierarchical analysis ofvariance and residual maximum likelihood. Computers & Geosciences, 2006, 32(9): 1320-1333.

[77] Arbia G, Lafratta G. Anisotropic spatial sampling designs for urban pollution. Journal of the Royal Statistical Society (Series C): Applied Statistics, 2002, 51: 223-234.

[78] Brus D J, Jansen M J W, Gruijter J J D. Optimizing two- and three-stage designs for spatial inventories of natural resources by simulated annealing. Environmental and Ecological Statistics, 2002, 9(1): 71-88.

[79] Bueso M C et al. A study on sensitivity of spatial sampling designs to a priori discretization schemes. Environmental Modelling & Software, 2005, 20(7): 891-902.

[80] Chao C T, Thompson S K. Optimal adaptive selection of sampling sites. Environmetrics, 2001, 12: 517-538.

[81] Van Groenigen J W, Pieters G, Stein A. Optimizing spatial sampling for multivariate contamination in urban areas. Environmetrics, 2000, 11(2): 227-244.

[82] Van Groenigen J W, Siderius W, Stein A. Constrained optimisation of soil sampling for minimisation of the kriging variance. Geoderma, 1999, 87: 239-259.

[83] Hoeting J A et al. Model selection for geostatistical models. Ecological Applications, 2006, 16(1): 87-98.

[84] Lin Y P, Rouhani S. Multiple-point variance analysis for optimal adjustment of a monitoring network. Environmental Monitoring and Assessment, 2001, 69(3): 239-266.

[85] Rogerson P A, Delmelle E, Batta R. Optimal sampling design for variables with varying spatial importance. Geographical Analysis, 2004, 36(2): 177-194.

[86] Saito H et al. Geostatistical interpolation of object counts collected from multiple strip transects: Ordinary Kriging versus finite domain kriging. Stochastic Environmental Research and Risk Assessment, 2005, 19(1): 71-85.

[87] Simbahan G C, Dobermaan A. Sampling optimization based on secondary information and its utilization in soil carbon mapping. Geoderma, 2006, 133: 345-362.

[88] Wiens D P. Robustness in spatial studies II: Minimax design. Environmetrics, 2005, 16(2): 205-217.

[89] Papritz A, Webster R. Estimating temporal change in soil monitoring: 2. Sampling from simulated fields. European Journal of Soil Science, 1995, 46(1): 13-27.

[90] Journel A G. Nonparametric geostatistics for risk and additional sampling assessment. In: Principles of Environmental Sampling. American Chemical Society, 1988: 45-72.

[91] Van Groenigen J W, Stein A. Constrained optimization of spatial sampling using continuous simulated annealing. Journal of Environmental Quality, 1998, 27: 1078-1086.

[92] Stevens Jr D L. Variable density grid-based sampling designs for continuous spatial populations. Environmetrics, 1997, 8: 167-195.

[93] Thompson S K. Factors infuencing the efficiency of adaptive cluster sampling, in technical report 94-0301. Technical Reports and Reprint Series. Center for Statistical Ecology and Environmental Statistics, The Pennsylvania State University, University Park, PA, 1994.

[94] Smith D R, Conroy M J, Brakhage D H. Efficiency of adaptive cluster sampling for estimating density of wintering waterfowl. Biometrics, 1995, 51: 777-788.

[95] Roesch F A Jr. Adaptive cluster sampling for forest inventories. Forest Science, 1993, 39: 655-669.

[96] Thompson S K. Adaptive cluster sampling based on order statistics. Environmetrics, 1996, 7: 123-133

[97] van Groenigen J W, Pieters G, Stein A. Optimizing spatial sampling for multivariate contamination in urban areas. Environmetrics, 2000, 11(2): 227-244.

[98] Zimmerman D L, Holland D M. Complementary co-kriging: Spatial prediction using data combined from several environmental monitoring networks Environmetrics, 2005, 16(3): 219-234.

[99] Bertolino F, Luciano A, Racugno W. Some aspects of detection networks optimization with the kriging procedure. Metron, 1983, 41(3): 91-107.

[100] Cattle J A, McBratney A B, Minasny B. Kriging method evaluation for assessing the spatial distribution of urban soil lead contamination. Journal of Environmental Quality, 2002, 31(5): 1576-1588.

[101] Chica-Olmo M, Luque-Espinar J A. Applications of the local estimation of the probability distribution function in environmental sciences by kriging methods. Inverse Problems, 2002, 18(1): 25-36.

[102] Emery X. A disjunctive Kriging program for assessing point-support conditional distributions. Computers & Geosciences, 2006, 32(7): 965-983.

[103] Fuentes M. A high frequency Kriging approach for non-stationary environmental processes. Environmetrics, 2001, 12 (2): 469-483.

[104] Lyon S W et al. Defining probability of saturation with indicator kriging on hard and soft data. Advances in Water Resources, 2006, 29(2): 181-193.

[105] Muller T G et al. Map quality for ordinary Kriging and inverse distance weighted interpolation. Nutrient Management& Soil & Plant Analysis, 2004, 68: 2042-2047.

[106] Triantafilis J et al. Mapping of salinity risk in the lower Namoi valley using non-linear kriging methods. Agricultural Water Management, 2004, 69(3): 203-229.

[107] Wang X J et al. Kriging and PAH pollution assessment in the topsoil of Tianjin area. Bulletin of Environmental Contamination and Toxicology, 2003, 71(1): 189-195.

[108] Yeh M S, Lin Y P, Chang L C. Designing an optimal multivariate geostatistical groundwater quality monitoring network using factorial kriging and genetic algorithms. Environmental Geology, 2006, 50(1): 101-121.

[109] Warrick A W, Myers D E. Optimization of sampling locations for variogram calculations. Water Resources Research, 1987, 23(3): 496-500.

[110] Groenigen J W. Spatial simulated annealing for optimizing sampling. In: Soares A, Gomez-Hernandez J, Froidevaux R (eds). GeoENV I: Geostatistics for Environmental Applications. Lisbon, Portugal: Kluwer Academic Publishers, 1997. 351-361.

[111] Zio S D, Fontanella L, Ippoliti L. Optimal spatial sampling schemes for environmental surveys Environmental and Ecological Statistics, 2004, 11(4): 397-414.

[112] Hastings H M, Sugihara G. Fractals: A User's Guide for the Natural Sciences. Oxford University Press, 1993.