Development of the kriging method with application. (English) Zbl 1091.62097

The real airborne data on the environmental gamma radioactivity in two areas of United Kingdom inspired this paper summarizing a part of the author’s PhD Thesis. The considered problem is to estimate the total radioactivity inventory on the examined area, i.e., the definite integral of a non-stationary random function \(Z(s)\) from its values in locations \(s_i,\;i=1,\dots , n.\) The procedure of the optimal linear prediction, known as universal kriging, was modified and used for the considered data. Because of its complexity (unknown non-constant trend \(\Lambda (s)\), different mean values in particular subareas, etc.), the data had to be first suitably stabilized; the best results have been obtained by the square-root transformation \(Y(s_i)=\sqrt {Z(s_i)}\) and then the kriging method was applied under the assumption that the residual random process \(U(s)=Y(s)-\Lambda (s)\) is a Gaussian second order stationary spatial process.
The obtained results were compared with two simple estimates of the total inventory (based on the regridding method consisting in the recalculation of the data to a regular lattice and on the observed mean value of the radioactivity) and gave comparable values. However, only the kriging method provides an estimator of the confidence interval for the prediction.
Reviewer: Ivan Saxl (Praha)


62M30 Inference from spatial processes
62P12 Applications of statistics to environmental and related topics
62M20 Inference from stochastic processes and prediction
Full Text: DOI EuDML


[1] M. Asli, D. Marcotte: Comparison of approaches to spatial estimation in a bivariate context. Mathematical Geology 27 (1995), 641-658.
[2] N. A. C. Cressie: Statistic for Spatial Data. Wiley, New York, 1993.
[3] P. J. Diggle, R. A. Moyeed and J. A. Tawn: Non-gaussian geostatistics. Research note, Lancaster University, Technical Report MA95 (1996).
[4] P. J. Diggle, J. A. Tawn and R. A. Moyeed: Model-based geostatistics. J. Roy. Statist. Soc. Ser. C 47 (1998), 199-350. · Zbl 0904.62119
[5] U. C. Herzfeld, D. F. Merriam: Optimization techniques for integrating spatial data. Mathematical Geology 27 (1995), 559-588.
[6] P. Krejčíř: The Theory and Applications of Spatial Statistics and Stochastic Geometry. PhD thesis, Charles University, Prague (2000).
[7] R. J. Serfling: Approximation Theorems of Mathematical Statistics. Wiley, New York, 1980. · Zbl 0538.62002
[8] M. L. Stein: Predicting integrals of random fields using observation an a lattice. Ann. Statist. 23 (1995), 1975-1990. · Zbl 0856.62083
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