Dr F P Agterberg, President, International Association for Mathematical Geology, has a few problems. The least of his problems is to change IAMG’s current name to International Association for Mathematical Geosciences. To bring the distance-weighted average and its central limit theorem back together again is just as pressing a problem as it is to count the degrees of freedom for a set and for the ordered set. So, I’ll try to put in a chronological context the cases of the missing variances and of the unwelcome degrees of freedom.
Agterberg talked about Autocorrelation Functions in Geology at the 1970 geostatistics colloquium in the USA. He had found some kind of “geologic prediction problem”, and drew a picture of it in Figure 1 of his paper. The same figure was reborn as “a typical kriging problem” in Figure 64 of his 1974 Geomathematics. As such, the same figure is published in the 1970 Colloquium Proceedings and in his 1974 Geomathematics. Why was a “geologic prediction problem” reborn as a “typical kriging problem”? I’ve studied the tortuous nomenclature of geostatistics and tried to figure out who lost what and when.
What both figures do have in common are symbols instead of Agterberg’s “known values” for the set of five irregularly spaced points. As luck would have it, the same function does apply to “a geologic prediction problem” and “a typical kriging problem.” Nomenclature has never been a strong suit in Matheron’s new science of geostatistics by symbols. David bragged in 1977, “It has been known for a long time that geostatisticians seem to have that capacity to change notations twice or more on the same page and still understand each other.” Not similarly blessed I’m guided by context and ISO symbols and terms. What I have known for twenty years is that geostatisticians have never derived the central limit theorem for the central value of a set of measured values with variable weights, and have never counted degrees of freedom for sets of measured values or ordered sets of measured values.
Agterberg did not mention that his 1970 and 1974 functions are one and the same. The correct contextual description of Agterberg’s function is “the distance-weighted average” of a set of five (5) measured values determined in samples selected at positions with different coordinates. He did not mention that both functions converge on the arithmetic mean as “irregularly spaced known values” become equidistant to P0. In his textbook, he does refer to the central limit theorem in Chapter 6 Probability and Statistics and in Chapter 7 Frequency Distributions and Functions of Independent Random Variables”. And he does refer to degrees of freedom in Chapter 6 Probability and Statistics and in Chapter 8 Statistical Dependence; Multiple Regression. He claimed in the second paragraph of Chapter 10 Stationary Random Variables and Kriging of his textbook, “The results can be used for interpolation and extrapolation.” What was Agterberg thinking?
Clark’s 1979 Practical Geostatistics was the first textbook to work with hypothetical uranium concentrations. That gave some touch and feel of real data but the set didn’t display spatial dependence. The author did study real Fisherian statistics where it was born but got into hanging out with the wrong crowd, and into worrying whether or not, ‘the Central Limit Theorem holds.” The good news is this theorem is bound to hold until the end of time! The bad news is IAMG’s President and his cronies on IAMG’s Council think it’s too late to bring central values and central limit theorems back together again.
Agterberg forgot to mention that his set of irregularly spaced points defines an infinite set of “predicted values” within this sample space and beyond it. He didn’t show how to test for spatial dependence by applying Fisher’s F-test to the variance of the set and the first variance term of the ordered set. He didn’t talk about a systematic walk to derive the variance of his ordered set. Agterberg knows his predicted value is a zero-dimensional point grade. If he were to agree that each zero-dimensional point grade does have its own variance, then he ought to revise his 1974 Geomathematics not only by deleting Chapter 10 Stationary Random Variables and Kriging but also by adding a chapter on precision and bias for mineral reserves and resources. That might be useful if the world’s mining industry were ever ready to set up and support an ISO Technical Committee for mineral reserve and resource estimation.
Agterberg’s 1974 Geomathematics, Mathematical Background and Geo-Science Applications, is a comprehensive textbook on the application of the queen of sciences in earth sciences. Agterberg covered much of the range of tools and techniques that mathematics provides in such rich abundance. This is why most of it will stand the test of time. In spite of that, Chapter 10 is bound to crumble under scrutiny because Agterberg’s geostatistical thinking was just as wrong as that of Matheron and his minions. Don’t take my word for it! Show Agterberg’s figure to a professor of mathematics. Ask her or him to explain whether or not each distance-weighted average has its own variance. And walk away if such a simple question about the Central Limit Theorem draws a blank!