A study on kriging small blocks

Dr Margaret Armstrong and Mr Normand Champigny had put this study on paper when they were toiling at the Centre de Géostatistique at Fontainebleau, France. Professor Dr Georges Matheron himself may have inspired them to compile their study in a paper. Be that as it may, this simple study was never added to Matheron’s magnum opus. It was kriging small blocks that inspired Armstrong and Champigny to elaborate on what they had detected. “Mine planners tended to define ore/waste limits as finely as possible”.

How about that? Perfect people are hard to find. So, the average mine planner was often tempted to over-smooth small blocks. The central tenet of this study was that over-smoothed estimates should not be used to derive recoverable reserves. That sort of research may well be the reason why Normand Champigny was awarded a Diploma in Geostatistics.

And why was this study published in CIM Bulletin, March 1989? Here’s why! Professor Dr Michel David and Professor Dr Alastair J Sinclair, PEng, PGeo, reviewed each and every paper in which kriging popped up in those days. Dr Frederik F Agterberg, Associate Editor with CIM Bulletin would not have hesitated to approve Armstrong and Champigny’s study. Here are a few facts and figures that neither the authors nor the reviewers knew about.

Dr Isobel Clark, in Chapter 4 Estimation of her 1979 Practical Geostatistics, derives not only the distance-weighted average AKA kriged estimate but also its variance. What she didn’t do was test for spatial dependence in the sample space defined by her hypothetical uranium concentrations. She sets the stage on page 3 of Chapter 1 Introduction under Figure 1.1. Hypothetical sampling and estimation situation. On page 5 she puts forward “the convenient assumption that there is no trend within the scale in which we are interested…” On the same page she fiddled with the factor 2 for “mathematical convenience” and fumbled her fickle “semi-variogram”. What went missing in her Index on page 127 above Disjunctive Kriging is Degrees of freedom. Dr Isobel Clark credits Professor Dr Andre Jounel and others at Fontainebleau who taught her all she knows “about the theory of the Theory of Regionalised Variables”. What a pity that Dr Clark did not know how to test for spatial dependence within the sample space defined by her set of hypothetical uranium concentrations. But then neither did any geostatistical reviewer for CIM Bulletin know how to test for spatial dependence between measured values in ordered sets!

Dr Isobel Clark, author of Practical Geostatistics
BSc, MSc, DIC, PhD, FSS, FSAIMM, FIMMM, CEng

Bringing Matheron’s new science of geostatistics to the world was quite a tour de force. Scores of geologists thought it odd that so much could be done with so few boreholes. But too few knew applied statistics well enough to figure out what was wrong with geostatistics. What Matheron and his disciples had failed to grasp was not only that all functions do have variances but also that sets of measured values do give degrees of freedom. That’s about all it took to do so much with so few boreholes! CIMMP’s archive has what Matheron’s magnum opus does not have. And that’s an authentic copy of Armstrong and Champigny’s study for a mere C$20.00.


What irked was that CIM Bulletin rejected Precision Estimates for Ore Reserves. I had mailed on September 28, 1989 four (4) copies to The Editor of CIM Publications. Not surprisingly, peer review of a paper that is at variance with the central tenets of geostatistical thinking turned out to be a blatantly biased and shamelessly self-serving sham. Our peers at CIM Bulletin were Professor Dr Michel David (1945-2000) and UBC Emeritus Professor Dr Alastair J Sinclair, PEng, PGeo. What a shame that unbiased confidence limits for metal contents and grades of ore reserves remain as rare as hen’s teeth.