Neither Agterberg nor de Wijs knew in 1958 that Dr Jan Visman’s 1947 PhD thesis on coal sampling was on file at TU Delft. I, too, was unaware of Visman’s work when I was a teaching assistant and a mature student in the early 1960s. I had been chief chemist with Dr Verwey but wanted to know more about sampling and statistics. The exchange of test samples and test results between trading partners was a thankless task to say the least. I knew all about the analytical variance but didn’t know how to estimate the variance of the primary sampling stage. I thought TU Delft would teach me what I wanted to know about sampling and statistics. One professor was H J de Wijs and the other a coal mining engineer. Neither knew of Visman’s work or of the properties of variances. In fact, H J de Wijs was Rector Magnificus when a student of his defended in 1965 a thesis in which transformation matrices played a key role. So, I left TU Delft, went to work for SGS in the Port of Rotterdam, and found out in 1967 about Visman’s 1947 sampling theory. So, Agterberg and I knew that mathematical statistics was shortchanged at TU Delft in those days. I don’t know why matrix and vector analysis were taught but sampling and statistics were ignored.
Dr Frederik P Agterberg had all but forgotten in this century what he had taught in 1958 at TU Delft. Here’s ad verbatim the very first paragraph of his eulogy, “Professor Georges Matheron (1930-2000) made fundamental contributions to science by establishing new theoretical frameworks in spatial statistics, random sets, mathematical morphology and the physics of random media”. Matheron was a French geologist who was called creator of geostatistics and founder of spatial statistics. I would never have praised Matheron’s surreal geostatistics let alone Journel's assumed spatial dependence. Whatever small minute contribution Matheron has made to science fell far short of real statistics. Surely, it did add up to surreal geostatistics. How ironic that he never got around to testing for spatial dependence between measured values in ordered sets. Why then did Dr Frederick P Agterberg see fit to praise Matheron’s work? He is Emeritus Scientist with the Geological Survey of Canada. Most of his 1974 textbook on Geomathematics has passed the test of time. And most of it will last much longer than Matheron’s magnum opus. What a shame that real statistics behind his 1970 and 1974 figures crumbles under scrutiny.
Figure 1 - Geologic prediction problem in 1970
Figure 64 - Typical kriging problem in 1974
Figure 64 - Typical kriging problem in 1974
Agterberg solved his geologic prediction problem by linear prediction in time series and assuming a two-dimensional autocorrelation function between his set of five (5) points. He has to explain how it came to pass that a geologic prediction problem in 1970 turned into a typical kriging problem by 1974. That was the very year that Elsevier published Agterberg’s Geomathematics. Cramped between its covers are some 600 pages of mostly real statistics, a lot of sound mathematics and a dash of Matheron’s surreal geostatistics. But why did Agterberg add Stationary random variables and kriging to an otherwise readable textbook?
What I see in each figure is a set of five (5) measured values in a two-dimensional sample space. If each of Agterberg’s points were equidistant to Po, then the central value of his set would be its arithmetic mean. David's famous Central Limit Theorem defines the functional relationship between a set of measured values with identical weights and its central value.
Agterberg refers to the Central Limit Theorem on pages 166, 206, 207 and 231. The number of degrees of freedom is a positive integer for a set of measured values with identical weights but a positive irrational for a set of measured values with variable weights. Agterberg refers to degrees of freedom on pages 174, 190 and 254. The Central Limit Theorem and the concept of degrees of freedom do not play a role in Chapter 10 - Stationary random variables and kriging. Dr Frederik P Agterberg, the author of Geomathematics and Emeritus Scientist with Natural Resources Canada, ought to explain why his Central Limit Theorem is not blessed with a variance and why his set of measured values is not blessed with degrees of freedom.