Abstract
Lognormal spatial data are common in mining and soil-science applications. Modeling the underlying spatial process as normal on the log scale is sensible; point kriging allows the whole region of interest to be mapped. However, mining and precision agriculture is carried out selectively and is based on block averages of the process on the original scale. Finding spatial predictions of the blocks assuming a lognormal spatial process has a long history in geostatistics. In this article, we make the case that a particular method for block prediction, overlooked in past times of low computing power, deserves to be reconsidered. In fact, for known mean, it is optimal. We also consider the predictor based on the “law” of permanence of lognormality. Mean squared prediction errors of both are derived and compared both theoretically and via simulation; the predictor based on the permanence-of-lognormality assumption is seen to be less efficient. Our methodology is applied to block kriging of phosphorus to guide precision-agriculture treatment of soil on Broom's Barn Farm, UK.
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References
Aitchison, J., and Brown, J. A. C., 1957, The lognormal distribution: Cambridge University Press, Cambridge, 176 p.
Aldworth, J., and Cressie, N., 1999, Sampling designs and prediction methods for Gaussian spatial processes, in Ghosh, S., ed., Multivariate Design and Sampling: Marcel Dekker, New York, p. 1–54.
Brown, G., and Sanders, J. W., 1981, Lognormal genesis: J. Appl. Probabil., v. 18, p. 542–547.
Cressie, N., 1985, Fitting variogram models by weighted least squares: Math. Geol., v. 17, p. 563– 586.
Cressie, N., 1993, Statistics for spatial data (revised edition): Wiley, New York, 900 p.
Cressie, N., and Pavlicova, M., 2002, Calibrated spatial moving average simulations: Stat. Model., v. 2, p. 267–279.
Cressie, N., and Pavlicova, M., 2005, Permanence of lognormality: Bias adjustment and kriging variances: in Leuangthong, O., and Deutsch, C. V., eds., Geostatistics Banff 2004: Springer, Dordrecht, p. 1027–1036.
de Wijs, H. J., 1951, Statistics of ore distribution. I: Frequency distribution of assay values: J. R. Netherlands Geol. Min. Soc., v. 13, p. 365–375.
Dowd, P. A., 1982, Lognormal kriging—The general case: Math. Geol., v. 14, p. 475–499.
Journel, A. G., 1980, The lognormal approach to predicting local distributions of selective mining unit grades: Math. Geol., v. 12, p. 285–303.
Marechal, A., 1974, Krigeage normal et lognormal. Centre de Morphologie Mathematique, Ecole des Mines de Paris, Publication N-376, 10 p.
Matheron, G., 1962, Traite de geostatistique appliquee: Memoires du Bureau de Recherche Geologiques et Minieres, no. 14, Ed. Technip, Tome 1, 333 p.
Matheron G., 1974, Effet proportionnel et lognormalite ou: Le retour du serpent de mer. Centre de Morphologie Mathematique, Ecole des Mines de Paris, Publication N-374, 43 p.
Rendu, J.-M., 1979, Normal and lognormal estimation: Math. Geol., v. 11, p. 407–422.
Rivoirard, J., 1990, A review of lognormal estimators for in situ reserves: Math. Geol., v. 22, p. 213–221.
Roth, C., 1988, Is lognormal kriging suitable for local estimation? Math. Geol., v. 30, p. 999– 1009.
Thorin, O., 1977, On the infinite divisibility of the lognormal distribution: Scand. Actuar. J., v. 1977, p. 121–148.
Webster, R., 2001, Statistics to support soil research and their presentation: Eur. J. Soil Sci., v. 52, 331–340.
Webster, R., and McBratney, A. B., 1987, Mapping soil fertility at Broom's Barn by simple kriging: J. Sci. Food Agric., v. 38, p. 97–115.
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Cressie, N. Block Kriging for Lognormal Spatial Processes. Math Geol 38, 413–443 (2006). https://doi.org/10.1007/s11004-005-9022-8
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DOI: https://doi.org/10.1007/s11004-005-9022-8