Sunday, October 18, 2009

Spatial dependence in mineral exploration

Some twenty years ago my son and I submitted to CIM Bulletin a paper on Precision Estimates for Ore Reserves. David, CIM Bulletin's reviewer, blew a fuse because we didn’t refer to “twenty years of geostatistical literature”. We did study David’s 1977 Geostatistical Ore Reserve Estimation and Clark’s 1979 Practical Geostatistics. Neither author showed how to test for spatial dependence. So, we showed how to test for spatial dependence between gold assays determined in bulk samples taken from twelve (12) rounds in a drift. CIM Bulletin was but one of several journals to reject our paper. Yet, the very same paper was praised by and published in Erzmetall 44, October 1991. We could not show how to estimate the intrinsic variance of gold because but a single bulk sample was taken from each round.
It was easy to estimate the intrinsic variance of gold in Bre-X’s phantom resource. Bre-X’s quality control program was based on selecting and testing duplicate test portions of every tenth crushed and salted core sample. The set of duplicate gold assays for Bre-X’s bonanza borehole BSSE198 gave enough degrees of freedom to estimate the analytical variance with a high degree of precision. Fisher’s F-test proved that the analytical variance and the first variance term of the ordered set are statistically identical. Hence, the intrinsic variance of gold in BSSE198 was statistically identical to zero. Plenty of placer gold was present in crushed and salted core samples but Bre-X’s bonanza borehole BSSE198 was barren.

When APCOM 2009 asked for abstracts, I talked to my son about presenting one more paper on our home turf. His talk about EMF at some school of mines in Nantes, France, took him too far away from Vancouver to attend APCOM 2009. Our abstract was based on a bulk sampling program at the Cerattepe project in Turkey where core recovery was poor. So, I advised my client to implement an interleaved bulk sampling program in order to derive unbiased confidence limits for in-situ gold and silver. Our abstract was accepted and Metrology in Mineral Exploration was approved.
I spoke to a small group on Thursday, October 8, 2009, at 15:30. I showed how to unscramble the Bre-X fraud, and how to derive the statistics for Cerattepe's bulk sampling program.

Spatial dependence is significant at 99.9% probability
Lag of 4.30 m at 95% probability is defined for gold

Spatial dependence is significant at 99.9% probability
Lag of 4.09 m at 95% probability is defined for silver

I explained how to correct those sampling variogram for the extraneous measurement variance estimated from pairs of interleaved primary samples, and how to derive 95% confidence limits for in-situ masses of gold and silver.
I asked my audience why the variance of Agterberg’s distance-weighted average point grade is still missing. I didn't get any response. Not a single question was asked. There was but a pinch of polite applause. My soul mate got an anonymous note together with the second coming of Clark’s 1979 Practical Geostatistics on DVD. Which APCOM 2009 sponsor ignored my question but did hand my spouse that anonymous note? Was it Gemcom? Or did Geovariances do it?
I was tickled pink with that priceless gift. In her first coming Clark cooked up a semi-variogram, berated those who "sloppily" call it a variogram. Yet, Clark praised Journal and his buddies for teaching her all she knows about “the theory of the Theory of Regionalized Variables.” Journel may well have taught Clark how to assume spatial dependence between measured values in ordered sets. He might even have cautioned Clark, too, not to become “too encumbered with Fischerian [sic!] statistics”. But what did Professor Dr William V Harper teach Dr Isobel Clark between 1979 and 2000? Sadly, Clark’s learning curve simply flat lined! She still doesn’t test for spatial dependence in sampling units and sample spaces. She still scolds those who work with variograms rather than with her own sacred semi-variogram. There's still no progress!
Statistics or geostatistics? Sampling error or nugget effect? Clark talked about those questions at WCSB4 in Cape Town on 21-23 October 2009. Sampling error adds a nice touch of Gy’ological thinking to Clark’s repertoire. Testing for spatial dependence failed to make her grade. Why did she take the factor two (2) out of degrees of freedom for ordered sets. Why does she deem too sloppy sampling variograms that show where orderliness in sample spaces or sampling units dissipates into randomness. Clark and Harper are ready to take Gy's sampling theory to sampling practices in mineral exploration, mining, processing, smelting and refining? Why does Harper not recognize that geostatistics is a scientific fraud? Strip the variance of the distance-weighted average, assume spatial dependence between measured values, interpolate by kriging, smooth the least biased subset of some infinite set of distance-weighted averages, and rig the rules of real statistics with impunity.