Geoscientists do not count degrees of freedom quite as well as do statisticians. Way too many have been taught some sort of new of science where spatial dependence need not be verified but may be assumed. Many geoscientists do not grasp why the properties of variances and the concept of degrees of freedom cannot be ignored with impunity. All the same, the world’s mining industry accepted this substitute for applied statistics because it does work miracles with a few boreholes drilled some distance apart.
Professor Dr Georges Matheron (1930-2000) developed his new science at his Centre de Géostatistique (CG) in Fontainebleau, France. The CG has posted much of his seminal work with its On-Line Library. I unscrambled it in Sampling and Statistics Explained, Chapter 2 Sampling Theory. In his 1954 Note Statistique No 3, Matheron derived the length-weighted average grade of a set of core samples with variable lengths but did not derive the variance of this weighted average grade. In his 1960 Note Géostatistique No 28, Matheron derived the length-weighted average grade of a block of in-situ ore but did not derive the variance of this weighted average grade either. Matheron made up the eponym “krigeage” in this 1960 paper to honor D G Krige for his work with weighted average gold grades at the Witwatersrand reef complex in South Africa.
Matheron’s “estimateur” in his 1960 paper turned out to be the length-weighted average grade of a three-dimensional block. Professor Dr Michel David (1945-2000), the author of the first textbook on geostatistics, found infinite sets of kriged estimates. Journel and Huijbregts, the authors of the second textbook, determined that infinite sets of kriged estimates give zero kriging variances. As luck would have it, each infinite set of zero-dimensional points fits in any three-dimensional block. Unluckily, neither the weighted average grade of a zero-dimensional point nor the weighted average grade of a single three-dimensional block does have a variance.
Functionally dependent values such as arithmetic means and weighted averages are not awarded degrees of freedom. By contrast, measured values do give degrees of freedom. It makes no sense whatsoever to test for spatial dependence in a sample space comprised of the first five positive integers of the infinite set. The probability to draw some subset of the infinite set of positive integers is just as immeasurable as the probability to select the least biased subset of an infinite set of kriged estimates. Geostatistians somehow beat such astronomical odds and select so-called BLUEs (Best Linear Unbiased Estimators) of infinite sets of kriged estimates.
Games of chance in the real world are quite different because we need to define some finite sample space of positive integers. For example, the ubiquitous 6/49 lottery is based on selecting a subset of six (6) integers from a sample space that consists of a subset of the positive integers 1, 2,…, 48, 49. The probability to win is P(win)=1/49·1/48·…·1/45·1/44≈ 0.000,000,00001 whereas the probability to lose is P(lose)=1–P(win)≈0.999,999,9999. Such probabilities are cast in stone because games of chance do give discrete outcomes. Unlike sampling practice where measured values are continuous and degrees of freedom are cast in stone. For example, Visman’s sampling experiment gives the arithmetic mean ash contents on dry basis for paired sets of small and large increments and the number of degrees of freedom. In symbols, a set of n measured values gives df=n–1, and the first term for the ordered set gives dfo=2(n–1). The number of degrees of freedom for a set of measured values with equal weights is a positive integer. For a set of measured values with variable weights, the number of degrees of freedom becomes a positive irrational.
In my 1984 textbook on Sampling and Weighing of Bulk Solids I explain how to derive the variance of the mass of metal in a quantity of mineral concentrate from its wet mass, moisture content and metal grade. This methodology requires realistic variance estimates for wet mass, moisture factor and grade factor. The interleaved sampling protocol gives realistic variance estimates at the lowest possible costs. ISO TC183 Copper, lead and zinc concentrates incorporated the methodology in CD13543–Determination of Mass of Contained Metal in the Lot. My son and I did not know the trouble we would see when we derived the variance of the mass of metal in a quantity of in-situ ore from its volume, density, and metal grade.