# Brownian motion along straight lines?

Professor G Matheron didn’t grasp in 1965 how to test for spatial dependence in sample spaces. What he did in 1970 is evoke Brownian motion along straight lines. He was scheduled to speak about it at the University of Kansas, Lawrence, Kentucky. He had taught A Marechal and J Serra all about kriging and smoothing at the Centre de Morphology Mathematique, Fontainebleau, France. Matheron has never explained why he stripped the variance off the distance-weighted average and called what was left a kriged estimate. Neither did The Founder of Spatial Statistics put in plain words why the distance-weighted average had metamorphosed into a kriged estimate. It was not D G Krige who called it a kriged estimate but Matheron! That’s in a nutshell why Professor Matheron could do so much with so little!!

Professor George Matheron had decided to talk about Random Functions and their Application in Geology. That’s when he talked about Brownian motion along straight lines. But a global estimation problem surfaced. Marechal and Serra did in Random Kriging in 1970 what M David would do in his 1977 Geostatistical Ore Reserve Estimation. In fact, M&S’s Figure 10 showed in 1970 what David’s Figure 203 did on page 286 of Chapter 10 The Practice of Kriging. One cannot possibly trust geostatistocrats who reject the Central Limit Theorem, the concept of degrees of freedom, and the power of Fisher’s F-test.

Dr F P Agterberg and his Autocorrelation Functions in Geology caught my attention when I perused his article. He was in those days no longer with the Kansas Geological Survey but with the Geological Survey of Canada. Figure 1 shows what Agterberg in 1970 had called:

Figure 1. Geologic prediction problem: values are known for five irregularly spaced points P1-P5. Value at P0 is unknown and to be estimated from five known values.

Fig. 64 Typical kriging problem; values are known at five points. Problem is to estimate value at point P0 from the known values at P1-P5.

This figure is also a take on Agterberg’s work. He is the author of Geomathematics, Mathematical Background and Geo-Science Applications. It was published in 1974 by Elsevier Scientific Publishing Company, Amsterdam New York. Dr F P Agterberg, in his 2000 tribute to Professor Dr G Matheron, remembered him as the Founder of Spatial Statistics. Yet, Matheron had never tested for spatial dependence between measured values in ordered sets. On the contrary, he was a master at assuming spatial dependence where it didn’t exist. Matheron could not have tested for spatial dependence at his Center de Morphology Mathematique. Why teach Fisher’s F-test if students are taught to assume spatial dependence? Agterberg’s tribute to Matheron taught me a lot about his thinking. When he was a graduate student at the University of Utrecht young Agterberg presented at the Technical University of Delft (TUDelft) a seminar on the skew frequency distribution of ore assay values. But that’s another part of my narrative!