Dr Frits P Agterberg is Emeritus Scientist with the Geological Survey of Canada. In the early 1990s he was but one of many scientists with NRCan’s precursor. Several scientists are members of Canadian Advisory Committees to the International Organization for Standardization. I have never met Agterberg at any such event. I derived a method that gives confidence limits for metal contents and grades of mined ores and mineral concentrates. It was approved as ISO DIS 13543, Determination of Mass of Contained Metal in the Lot. What I wanted to do was apply the same method to in-situ ores and coals. Cominco’s geologists taught me a bit about kriging and smoothing but I didn’t get the gist of it. A mining company gave me a set of gold assays for ordered rounds in a drift to play with.
My son and I studied a few geostat books. We found David’s 1977 Geostatistical Ore Reserve Estimation to be short on statistics and long on geostat drivel. David did pay tribute to the ‘famous’ Central Limit Theorem but didn’t take to working with it. Neither did he take to counting degrees of freedom. Degrees of freedom played a cameo role when he pointed to an earlier work of geologists. David didn’t derive confidence limits for metal grades and contents of ore deposits. He was right on cue when he wrote “…statisticians will find many unqualified statements…” David didn’t write he would throw a temper tantrum if any one dared to.
David was but one of a score of geostat thinkers who thrashed Precision Estimates for Ore Reserves and made a mockery of peer review in the process. But did they ever know how to stake out their own turf! They coined all sorts of terms and never stopped nattering nonsense neologisms among themselves. I’m not gifted in verbal discourse. It took me a while to grasp that kriged estimates, kriged estimators, estimated values, and simulated values, are birds of a feather in geostat speak. What did blow my mind were infinite sets of kriged estimates, zero kriging variances, and a dreadful disrespect for degrees of freedom. Who could have made up so much poppycock?
Agterberg made up quite a bit of it. He cooked up a distance-weighted average point grade that didn’t have a variance. He failed to put in plain words why his function lost its variance. Neither did he ever tell me why his zero-dimensional distance-weighted average point grade didn’t have a variance. It led me to guess that this lost variance wasn’t his proudest feat. What is still beyond Agterberg’s grasp in 2009 is one-to-one correspondence between functions and variances.
Matheron’s new science of geostatistics drifted across the Channel and the Atlantic Ocean and made landfall on the North American continent in 1970. The mining industry was gung-ho to swallow least biased subsets of infinite sets of kriged estimates with hook, line and sinker. Kriging and smoothing sounded so soothing. How its practitioners could beat the odds of selecting least biased subsets of infinite sets of kriged estimates troubled but a few. The list of those who couldn’t care less would stack a Mining Hall of Shame. I got to the bottom of Matheron’s odd statistics long before the Bre-X fraud. But no one cared!
I took my time to find out who lost what, when and where. It was Agterberg who brought to light a typical geologic prediction problem in 1970. I took a look and saw a distance-weighted average. He found a typical kriging problem in 1974. All I saw was the same distance-weighted average.
Here are a few of Agterberg’s real problems. He didn’t know how to test for spatial dependence between his ordered point grades. He didn’t know how to derive the variance of his distance-weighted average point grade. He didn’t know how to count degrees of freedom either for the set or for the ordered set. Yet, Agterberg does point to degrees of freedom on pages 174, 190 and 254 of his 1974 Geomathematics.
What’s more, Agterberg didn’t take to door-to-door sales walks. Such a walk would visit each point only once and cover the shortest possible distance between all points. He could then have applied Fisher’s F-test to the variance of the set and the first variance term of the ordered set. Agterberg himself did apply Fisher’s F-test on page 187 of his 1974 Geomathematics. And he does refer to Sir Ronald A Fisher‘s work on nine (9) pages!
Agterberg does not refer to Dr Jan Visman’s work. Visman was a Dutch coal mining engineer who worked with the Dutch State Mines during the war. His PhD thesis proved the variance of the primary sample selection stage to be the sum of the composition variance and the distribution variance. Visman immigrated to Canada in 1951 and worked with the Department of Mines and Technical Surveys in Ottawa. He wrote Towards a Common Basis for the Sampling of Materials (Research Report R 93, July 1962). The advent of ash analyzers for coal and on-stream analyzers for slurry flows led to a fundamental understanding of what spatial dependence between measured values in ordered sets is all about. Yet, spatial dependence in sampling units and sample spaces stayed as profound a mystery to the geostatocracy as were the properties of variances.
Agterberg prefers oral criticism. Once upon a time he did reply in writing. On October 11, 2004, he called me …an iconoclast with respect to spatial statistics and kriging.” He insisted, “By now this approach is well established in mathematical statistics.” He got it all wrong again. Kriging and mathematical statistics have as little in common as alchemy and chemistry. What is ringing kriging’s bell is climate change. That’s were Agterberg’s zero-dimensional distance-weighted average will always have a variance. Whether he likes it or not!