Kriging Regionalized Positive Variables Revisited: Sample Space and Scale Considerations

Abstract Frequently, regionalized positive variables are treated by preliminarily applying a logarithm, and kriging estimates are back-transformed using classical formulae for the expectation of a lognormal random variable. This practice has several problems (lack of robustness, non-optimal confidence intervals, etc.), particularly when estimating block averages. Therefore, many practitioners take exponentials of the kriging estimates, although the final estimations are deemed as non-optimal. Another approach arises when the nature of the sample space and the scale of the data are considered. Since these concepts can be suitably captured by an Euclidean space structure, we may define an optimal kriging estimator for positive variables, with all properties analogous to those of linear geostatistical techniques, even for the estimation of block averages. In this particular case, no assumption on preservation of lognormality is needed. From a practical point of view, the proposed method coincides with the median estimator and offers theoretical ground to this extended practice. Thus, existing software and routines remain fully applicable.

[1]  J. Rendu,et al.  Normal and lognormal estimation , 1979 .

[2]  J. Aitchison,et al.  The Lognormal Distribution. , 1958 .

[3]  Ole A. Nielsen.  An Introduction to Integration and Measure Theory , 1997 .

[4]  J. Chilès,et al.  Geostatistics: Modeling Spatial Uncertainty , 1999 .

[5]  Noel A Cressie,et al.  Statistics for Spatial Data. , 1992 .

[6]  S. P. Neuman,et al.  Estimation of Aquifer Parameters Under Transient and Steady State Conditions: 1. Maximum Likelihood Method Incorporating Prior Information , 1986 .

[7]  On the consistency of the indirect lognormal correction , 2004 .

[8]  V. Pawlowsky-Glahn,et al.  Geometric approach to statistical analysis on the simplex , 2001 .

[9]  Eulogio Pardo-Igúzquiza,et al.  Variance–Covariance Matrix of the Experimental Variogram: Assessing Variogram Uncertainty , 2001 .

[10]  R. Olea,et al.  Geostatistical Analysis of Compositional Data , 2004 .

[11]  Clayton V. Deutsch,et al.  GSLIB: Geostatistical Software Library and User's Guide , 1993 .

[12]  Andre G. Journel,et al.  The lognormal approach to predicting local distributions of selective mining unit grades , 1980 .

[13]  Jacques Rivoirard.  A review of lognormal estimators forin situ reserves , 1990 .

[14]  G. Verly,et al.  The multigaussian approach and its applications to the estimation of local reserves , 1983 .

[15]  F. Pukelsheim.  The Three Sigma Rule , 1994 .

[16]  William V. Harper,et al.  Practical geostatistics 2000 , 2000 .

[17]  M. L. Eaton.  Multivariate statistics : a vector space approach , 1985 .

[18]  A Simple and Robust Lognormal Estimator , 1997 .

[19]  David Michel,et al.  Geostatistical Ore Reserve Estimation , 1977 .

[20]  D. Mcalister,et al.  XIII. The law of the geometric mean , 1879, Proceedings of the Royal Society of London.

[21]  Peter A. Dowd,et al.  Lognormal kriging—the general case , 1982 .

[22]  C. Roth.  Is Lognormal Kriging Suitable for Local Estimation? , 1998 .

[23]  Margaret Armstrong,et al.  Positive definiteness is not enough , 1992 .

[24]  D. Myers.  Matrix formulation of co-kriging , 1982 .