Missing variance

These equations derive from the variance of the general function as defined in probability theory. A kriged estimate is a functionally dependent variable of a set of (independently) measured values of a random variable determined at different coordinates in a sample space. Two or more measured values with different coordinates define an infinite set of kriged estimates. In classical statistics, one-on-one correspondence between central values and variances implies that each distance-weighted average in an infinite set has its own variance. In geostatistics, however, variances of kriged estimates vanished without a trace.

The central limit theorem describes the relationship between the variance of a set of independently measured values with equal weights and the variance of its arithmetic mean. In Geostatistical Ore Reserve Estimation, Chapter 2, page 33, David refers to "the famous central limit theorem". Both equations give the variance of all types of weighted averages but both turn into the central limit theorem for measured values with equal weights. Just the same, the equation for the variance of a single kriged estimate is nowhere to be found in the geostatistical literature!

I submitted in January 1993 an abstract for "The Properties of Variances" to Geostatistics for the Next Century, a forum at McGill University on June 3-5, 1993. The abstract made reference to the variance of a kriged estimate but the properties of variances as defined in probability theory and classical statistic failed to arouse geostatisticians who were poised to travel early beyond 2000. When I asked Dr Roussos Dimitrakopoulos, the Forum's Chairman and nowadays one of the Associate Editors with the Journal of Mathematical Geology, whether I could attend without presenting a paper, he didn't reply. It is indeed a fact that I have given JMG "more than its share of distracting invective" before I tried to crash the bash and interrupt their journey into the next century with a fundamentally flawed science. Ironically, geostatisticians could have detected Bre-X's salting scam if they would have been conversant with the properties of variances.