Probability and Statistics - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 45

(Krige 1951 ). This technique can be regarded as a first application of “kriging”,
which is a translation of the term “Krigeage” originally coined by Georges
Matheron ( 1962 ) who remarked that use of this word was sanctioned by the
[French] Commissariat ` l'Energie Atomique to honor work by Krige on the bias
affecting estimation of mining block grades from sampling in their surroundings,
and on the correction coefficients that should be applied to avoid this bias. Later,
Matheron ( 1967 ) urged the English-speaking community to adopt the term

“kriging” which now is used worldwide.
Krige's original paper was translated into the French and republished in 1955 in
a special issue of Annales des Mines on the use of mathematical statistics in
economic geology. It is followed by a paper by Matheron ( 1955 ) who emphasized
“permanence” of lognormality in that gold assays from smaller and larger blocks all
have lognormal frequency distributions with variances decreasing with increasing
block size. Matheron discusses “Krige's formula” for the propagation of variances
of logarithmically transformed mining assays, which states that the variance for
small blocks within a large block is equal to the variance for the small blocks within
intermediate-size blocks plus the variance of the intermediate-size blocks within
the large block. This empirical formula could not be reconciled with early theory of
mathematical statistics but it constitutes a characteristic feature in the spatial model
of orebodies developed in the late 1940s by the Dutch mining engineer Hans de
Wijs ( 1951 ) whose approach helped Matheron to formulate the idea of “regio-
nalized random variable”. Rather than using autocorrelation coefficients as were
generally employed in time series analysis under the assumption of existence of a
mean and finite variance, Matheron ( 1962 ) introduced the variogram as a basic tool
for structural analysis of spatial continuity of chemical element concentration
values in rocks and orebodies. This is because the variogram allows for the
possibility of infinitely large variance as would result from the de Wijsian
model for indefinitely increasing distances between sampling points. Aspects of
this model were adopted by Krige ( 1978 ) in his monograph Lognormal-de
Wijsian Geostatistics for Ore Evaluation summarizing earlier studies including
his successful application to characterize self-similar gold and uranium distribution
patterns in the Klerksdorp goldfield in South Africa (also see Sect. 11.1 ).
Initially, Georges Matheron performed his geostatistical work in the 1950s with
mining applications. In the late 1960s his approach caught the attention of mathe-
matical statisticians including Geof Watson and John Tukey. Noel Cressie ( 1991 ),
a former PhD student of Watson wrote the textbook Statistics for Spatial Data
casting Matheron's approach into a mathematical statistical context. At the Biennial
Session of the International Statistical Institute held in Seoul, 2001, Georges
Matheron, John Tukey and Lucien Le Cam jointly were honoured posthumously
as great mathematical statisticians from the second half of the twentieth Century.
It illustrates that the idea of “regionalized random variable” has become a corner
stone of mathematical statistics.
What is kriging variance and why do degrees of freedom not play an important
role in spatial statistics? Formulas for kriging variances can be found in all
geostatistical textbooks. In general, they are larger than variances that would be
estimated by making the simple assumption that the values used for kriging are