Two contrasting spatial processes with a common variogram: inference about spatial models from higher-order statistics
Geostatistical analysis of soil properties is undertaken to allow prediction of values of these properties over regions or at unsampled locations. A key step in geostatistical analysis is the estimation of a variogram function that describes the spatial covariance structure of the variable in questi...
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| Vydáno v: | European journal of soil science Ročník 61; číslo 4; s. 479 - 492 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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Oxford, UK
Blackwell Publishing Ltd
01.08.2010
Blackwell |
| Témata: | |
| ISSN: | 1351-0754, 1365-2389 |
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| Abstract | Geostatistical analysis of soil properties is undertaken to allow prediction of values of these properties over regions or at unsampled locations. A key step in geostatistical analysis is the estimation of a variogram function that describes the spatial covariance structure of the variable in question. If it can be assumed plausibly that the data are a realization of a second‐order stationary multivariate normal random function then this function is entirely characterized by its mean (expectation) and spatially dependent covariance. Because of this, the variogram is sometimes computed as a general ‘descriptor’ of spatial variation, and used, for example, to compare the spatial structure at within‐aggregate scales of soils under different management, or to compare soils from different land uses with respect to the spatial structure of their microbial populations. The objective of this paper is to draw attention to the limited value of the variogram for characterizing spatial variation (as opposed to deriving best linear unbiased predictions). Specifically, it is shown how two contrasting processes, one of which gives rise to a multivariate normal random function (a convolution filter applied to independent identically distributed random values) and one which does not (a partition of space into random sets), may have the same variogram function. A diagnostic is proposed that indicates which of these two processes is most plausible as a model for a data set. This will allow the spatial analysis of soil data to give greater insight into the factors underlying the variation of a soil property, and may permit more realistic simulation of soil properties. |
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| AbstractList | Geostatistical analysis of soil properties is undertaken to allow prediction of values of these properties over regions or at unsampled locations. A key step in geostatistical analysis is the estimation of a variogram function that describes the spatial covariance structure of the variable in question. If it can be assumed plausibly that the data are a realization of a second-order stationary multivariate normal random function then this function is entirely characterized by its mean (expectation) and spatially dependent covariance. Because of this, the variogram is sometimes computed as a general 'descriptor' of spatial variation, and used, for example, to compare the spatial structure at within-aggregate scales of soils under different management, or to compare soils from different land uses with respect to the spatial structure of their microbial populations. The objective of this paper is to draw attention to the limited value of the variogram for characterizing spatial variation (as opposed to deriving best linear unbiased predictions). Specifically, it is shown how two contrasting processes, one of which gives rise to a multivariate normal random function (a convolution filter applied to independent identically distributed random values) and one which does not (a partition of space into random sets), may have the same variogram function. A diagnostic is proposed that indicates which of these two processes is most plausible as a model for a data set. This will allow the spatial analysis of soil data to give greater insight into the factors underlying the variation of a soil property, and may permit more realistic simulation of soil properties. |
| Author | Lark, R. M. |
| Author_xml | – sequence: 1 givenname: R. M. surname: Lark fullname: Lark, R. M. email: murray.lark@bbsrc.ac.uk organization: Rothamsted Research, Harpenden, Hertfordshire AL5 2JQ, UK |
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| Cites_doi | 10.1111/j.1574-6941.2002.tb00996.x 10.1111/j.1365-2389.2004.00629.x 10.1016/0012-821X(95)00049-I 10.1016/j.geoderma.2005.11.008 10.1007/s00248-006-9062-8 10.1016/j.geoderma.2005.08.004 10.1016/S0022-1694(97)00056-5 10.1111/1467-9868.00269 10.1016/j.advwatres.2004.04.002 10.1023/A:1021765405952 10.1111/j.1365-2389.2009.01152.x 10.1093/oso/9780195115383.001.0001 10.1023/A:1014009426274 10.1080/00401706.1991.10484771 10.1016/S0016-7061(00)00036-7 10.1080/00401706.1967.10490438 10.1016/j.geoderma.2007.04.028 10.1016/j.envpol.2005.12.010 10.1111/j.1365-2389.1980.tb02084.x 10.1002/9780470517277 10.1007/BF01063317 10.1002/9781119115151 10.1111/j.1365-2389.1975.tb01942.x 10.1016/S0038-0717(97)00107-7 |
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| References | Rivoirard, J. 1994. Introduction to Disjunctive Kriging and Non-linear Geostatistics. Oxford University Press, Oxford. Stacey, K.F., Lark, R.M., Whitmore, A.P. & Milne, A.E. 2006. Using a process model and regression kriging to improve predictions of nitrous oxide emissions from soil. Geoderma, 135, 107-117. Nunan, N., Ritz, K., Rivers, M., Feeney, D.S. & Young, I.M. 2006. Investigating microbial micro-habitat structure using X-ray computed tomography. Geoderma, 133, 398-407. Pardo-Igúzquiza, E. 1998. Maximum likelihood estimation of spatial covariance parameters. Mathematical Geology, 30, 95-108. Lark, R.M. 2009. A stochastic-geometric model of soil variation. European Journal of Soil Science, 60, 706-719. Zimmerman, D.L. & Zimmerman, M.B. 1991. A comparison of spatial semivariogram estimators and corresponding ordinary kriging predictors. Technometrics, 33, 77-91. Matheron, G. 1962. Traité de Géostatistique Appliqué. Tome 1. Mémoires du Bureau de Recherches Géologiques et Minières, Paris. Webster, R. 2000. Is soil variation random? Geoderma, 97, 149-163. Lark, R.M., Milne, A.E., Addiscott, T.M., Goulding, K.W.T., Webster, C.P. & O'Flaherty, S. 2004. Scale- and location-dependent correlation of nitrous oxide emissions with soil properties: an analysis using wavelets. European Journal of Soil Science, 55, 601-610. Feeney, D.S., Crawford, J.W., Daniell, T., Hallett, P.D., Nunan, N., Ritz, K., Rivers, M. & Young, I.M. 2006. Three-dimensional microorganization of the soil-root-microbe system. Microbial Ecology, 52, 151-158. Cohen, A.C. 1967. Estimation in mixtures of two normal distributions. Technometrics, 9, 15-28. Minasny, B. & McBratney, A.B. 2007. Spatial prediction of soil properties using EBLUP with the Matérn covariance function. Geoderma, 140, 324-336. Allègre, C.J. & Lewin, E. 1995. Scaling laws and geochemical distributions. Earth & Planetary Science Letters, 132, 1-13. Rawlins, B.G., Lark, R.M., Webster, R. & O'Donnell, K.E. 2006. Historic metal deposition from atmospheric smelter emissions on Humberside, UK: 1. Magnitude and extent of contamination based on soil survey data. Environmental Pollution, 143, 416-426. Webster, R. & Oliver, M.A. 2007. Geostatistics for Environmental Scientists, 2nd edn. John Wiley & Sons, Chichester. Brown, P.E., Kåresen, K.F., Roberts, G.O. & Tonellato, S. 2000. Blur-generated non-separable space-time models. Journal of the Royal Statistical Society B, 62, 847-860. Webster, R. & Cuanalo de la C.H.E. 1975. Soil transect correlograms of north Oxfordshire and their interpretation. Journal of Soil Science, 26, 176-194. White, R.W., Haigh, R.A. & MacDuff, J.H. 1987. Frequency distributions and spatially dependent variability of ammonium and nitrate concentrations in soil under grazed and ungrazed grassland. Fertilizer Research, 11, 193-208. Franklin, R.B., Blum, L.K., McComb, A.C. & Mills, A.L. 2002. A geostatistical analysis of small-scale spatial variability in bacterial abundance and community structure in salt marsh creek bank sediments. FEMS Microbiology Ecology, 42, 71-80. Christakos, G. 2000. Modern Spatiotemporal Geostatistics. Oxford University Press, New York. Cressie, N.A.C. 1993. Statistics for Spatial Data, revised edn. Wiley, New York. Kolovos, A., Christakos, G., Hristopulos, D.T. & Serre, M.L. 2004. Methods for generating non-separable spatiotemporal covariance models with potential environmental applications. Advances in Water Resources, 27, 815-830. Burgess, T.M. & Webster, R. 1980. Optimal interpolation and isarithmic mapping of soil properties. I. The semivariogram and punctual kriging. Journal of Soil Science, 31, 315-331. Görres, J.H., Dichiaro, M.J., Lyons, J.B. & Amador, J.A. 1998. Spatial and temporal patterns of soil biological activity in a forest and an old field. Soil Biology & Biochemistry, 30, 219-230. Snedecor, G.W. & Cochran, W.G. 1989. Statistical Methods, 8th edn. Iowa State University Press, Ames, IA. Romano, N. & Santini, A. 1997. Effectiveness of using pedo-transfer functions to quantify the spatial variability of soil water retention characteristics. Journal of Hydrology, 202, 137-157. Strebelle, S. 2002. Conditional simulation of complex geological structures using multiple-point statistics. Mathematical Geology, 34, 1-21. IMSL 1997. IMSL, Fortran Routines for Mathematical Applications. Visual Numerics, Houston, TX. 1987; 11 2006; 52 2007; 140 2004; 27 2009; 60 1991; 33 2002; 34 1997 2007 1994 1993 1995; 132 2006; 133 2006; 135 2004; 55 1997; 202 1967; 9 1980; 31 2000 2002; 42 2000; 97 2000; 62 2006; 143 1975; 26 1962 1998; 30 1989 e_1_2_9_30_1 e_1_2_9_11_1 e_1_2_9_10_1 e_1_2_9_13_1 Christakos G. (e_1_2_9_5_1) 2000 e_1_2_9_15_1 e_1_2_9_14_1 e_1_2_9_17_1 Snedecor G.W. (e_1_2_9_23_1) 1989 e_1_2_9_19_1 e_1_2_9_18_1 e_1_2_9_20_1 Matheron G. (e_1_2_9_16_1) 1962 e_1_2_9_22_1 e_1_2_9_24_1 e_1_2_9_8_1 e_1_2_9_7_1 e_1_2_9_6_1 e_1_2_9_4_1 e_1_2_9_3_1 e_1_2_9_2_1 Rivoirard J. (e_1_2_9_21_1) 1994 IMSL (e_1_2_9_12_1) 1997 e_1_2_9_9_1 e_1_2_9_26_1 e_1_2_9_25_1 e_1_2_9_28_1 e_1_2_9_27_1 e_1_2_9_29_1 |
| References_xml | – reference: Lark, R.M., Milne, A.E., Addiscott, T.M., Goulding, K.W.T., Webster, C.P. & O'Flaherty, S. 2004. Scale- and location-dependent correlation of nitrous oxide emissions with soil properties: an analysis using wavelets. European Journal of Soil Science, 55, 601-610. – reference: Christakos, G. 2000. Modern Spatiotemporal Geostatistics. Oxford University Press, New York. – reference: Burgess, T.M. & Webster, R. 1980. Optimal interpolation and isarithmic mapping of soil properties. I. The semivariogram and punctual kriging. Journal of Soil Science, 31, 315-331. – reference: Zimmerman, D.L. & Zimmerman, M.B. 1991. A comparison of spatial semivariogram estimators and corresponding ordinary kriging predictors. Technometrics, 33, 77-91. – reference: IMSL 1997. IMSL, Fortran Routines for Mathematical Applications. Visual Numerics, Houston, TX. – reference: Kolovos, A., Christakos, G., Hristopulos, D.T. & Serre, M.L. 2004. Methods for generating non-separable spatiotemporal covariance models with potential environmental applications. Advances in Water Resources, 27, 815-830. – reference: Webster, R. 2000. Is soil variation random? Geoderma, 97, 149-163. – reference: Feeney, D.S., Crawford, J.W., Daniell, T., Hallett, P.D., Nunan, N., Ritz, K., Rivers, M. & Young, I.M. 2006. Three-dimensional microorganization of the soil-root-microbe system. Microbial Ecology, 52, 151-158. – reference: Matheron, G. 1962. Traité de Géostatistique Appliqué. Tome 1. Mémoires du Bureau de Recherches Géologiques et Minières, Paris. – reference: Strebelle, S. 2002. Conditional simulation of complex geological structures using multiple-point statistics. Mathematical Geology, 34, 1-21. – reference: Minasny, B. & McBratney, A.B. 2007. Spatial prediction of soil properties using EBLUP with the Matérn covariance function. Geoderma, 140, 324-336. – reference: Webster, R. & Oliver, M.A. 2007. Geostatistics for Environmental Scientists, 2nd edn. John Wiley & Sons, Chichester. – reference: Görres, J.H., Dichiaro, M.J., Lyons, J.B. & Amador, J.A. 1998. Spatial and temporal patterns of soil biological activity in a forest and an old field. Soil Biology & Biochemistry, 30, 219-230. – reference: White, R.W., Haigh, R.A. & MacDuff, J.H. 1987. Frequency distributions and spatially dependent variability of ammonium and nitrate concentrations in soil under grazed and ungrazed grassland. Fertilizer Research, 11, 193-208. – reference: Rawlins, B.G., Lark, R.M., Webster, R. & O'Donnell, K.E. 2006. Historic metal deposition from atmospheric smelter emissions on Humberside, UK: 1. Magnitude and extent of contamination based on soil survey data. Environmental Pollution, 143, 416-426. – reference: Allègre, C.J. & Lewin, E. 1995. Scaling laws and geochemical distributions. Earth & Planetary Science Letters, 132, 1-13. – reference: Brown, P.E., Kåresen, K.F., Roberts, G.O. & Tonellato, S. 2000. Blur-generated non-separable space-time models. Journal of the Royal Statistical Society B, 62, 847-860. – reference: Stacey, K.F., Lark, R.M., Whitmore, A.P. & Milne, A.E. 2006. Using a process model and regression kriging to improve predictions of nitrous oxide emissions from soil. Geoderma, 135, 107-117. – reference: Cressie, N.A.C. 1993. Statistics for Spatial Data, revised edn. Wiley, New York. – reference: Webster, R. & Cuanalo de la C.H.E. 1975. Soil transect correlograms of north Oxfordshire and their interpretation. Journal of Soil Science, 26, 176-194. – reference: Pardo-Igúzquiza, E. 1998. Maximum likelihood estimation of spatial covariance parameters. Mathematical Geology, 30, 95-108. – reference: Lark, R.M. 2009. A stochastic-geometric model of soil variation. European Journal of Soil Science, 60, 706-719. – reference: Snedecor, G.W. & Cochran, W.G. 1989. Statistical Methods, 8th edn. Iowa State University Press, Ames, IA. – reference: Franklin, R.B., Blum, L.K., McComb, A.C. & Mills, A.L. 2002. A geostatistical analysis of small-scale spatial variability in bacterial abundance and community structure in salt marsh creek bank sediments. FEMS Microbiology Ecology, 42, 71-80. – reference: Rivoirard, J. 1994. Introduction to Disjunctive Kriging and Non-linear Geostatistics. Oxford University Press, Oxford. – reference: Romano, N. & Santini, A. 1997. Effectiveness of using pedo-transfer functions to quantify the spatial variability of soil water retention characteristics. Journal of Hydrology, 202, 137-157. – reference: Cohen, A.C. 1967. Estimation in mixtures of two normal distributions. Technometrics, 9, 15-28. – reference: Nunan, N., Ritz, K., Rivers, M., Feeney, D.S. & Young, I.M. 2006. Investigating microbial micro-habitat structure using X-ray computed tomography. 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| SubjectTerms | Agronomy. Soil science and plant productions Biological and medical sciences covariance Earth sciences Earth, ocean, space Exact sciences and technology Fundamental and applied biological sciences. Psychology geostatistics land use population structure prediction soil analysis soil properties Soil science Soils Surficial geology |
| Title | Two contrasting spatial processes with a common variogram: inference about spatial models from higher-order statistics |
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