LETTER TO THE EDITOR — Kriging merely geostatistical ‘gibberish’

Several people have called my attention to articles and letters on geostatistics in recent issues of The Northern Miner and urged me to comment on them.

None of the authors of these articles has grasped the fundamental issue, which is that kriging, the so-called mathematical tool used by geostatisticians, gives ridiculous results that have nothing to do with weighing sample assays. The weights it produces are for random variables, a difficult concept from stochastic process theory; and nobody mines random variables — they mine assayed gold, copper, etc.

To be sure, when the weird results produced by kriging are averaged with tens or hundreds of others, they are hard to see, just as were the astronomical errors made for 1,500 years by Ptolemaic astronomers, who thought the sun went round the earth.

A crucial test of kriging can be performed by anyone using any of the many “canned” kriging programs available, or by writing the simple program himself.

Exhibit I shows a row of 13 circles on a straight line, designating points from which samples have been taken and assayed. To the left of each circle is the factor (in per cent) by which the assay of a sample at that point should be multiplied and added into a weighted average estimate of the assay at the estimate point. Also to the left is a bar chart showing these percentages which, in “ordinary kriging,” add to 100%.

What this exhibit says is that if we have a core 12 units long with a gold assay of 100 grams per tonne at one end point, while assays at the other 12 points are barren, then the estimated grade at the estimate point is 29.72 grams. If that same stringer happens to be one unit from the end, then the estimated grade is 3.16 grams.

I call this the “drill core paradox,” and it is a genuine mathematical problem — a mathematician would call it a counter-example disproving kriging. Is there a single reader of The Northern Miner who believes this totally absurd result?

Yet these percentages may be found by any reader using a computer program for “ordinary kriging” and the spherical covariogram, so beloved by krigers, with a range of 40 units.

Exhibit II shows what would happen if the sampler moved the sample point a half-unit closer to the estimate point to get around some obstruction at the centre point. If the sample at the centre point showed 100 grams gold and the rest were blank, the estimate would be 43.44 grams per tonne, a jump from 4.07 grams, caused by changing the distance to the estimate point from 12 to 11.5 units. A change of a mere 4% increases the weight by a factor of more than 10. However, if the 100-gram assay were either one unit above or below the centre, the estimated gold assay would be minus-14.44 grams.

Maybe it would be better for the sampler to move the centre sample location half a unit farther from the estimate point, as outlined in Exhibit III. If the centre assay is 100 grams per tonne and the rest are blank, the estimate is minus-30.38 grams per tonne. The weighing factor changes following a shift in distance to 12.5 units from 11.5 units, from 43.44% to minus-30.38%, or a total of 73.82%.

Krigers call this the “screen effect,” and its reasonable-sounding name apparently makes them stop worrying about it. The same is true of the “de-clustering effect” in Exhibit I, though they do talk about the effect of some “infinite domain” and propose manipulating sample locations or fudging the calculations to “fix” it.

But can these negative assays be so easily explained away?

Suppose that instead of assays, these were porosities in a geological formation proposed for isolating nuclear waste from water for the next 100,000 years; high porosity might well be fatal for many people.

But a high porosity at any point with a negative weight would actually decrease the weighted average porosity estimate. In Exhibit III, for example, it would be multiplied by 30.38% and subtracted from the sum of the products of the other factors.

Does anybody believe any of this? Of course not, once they have seen it.

The reason is that kriging simply is not telling us about weighting sample assays. It is telling us about weighting random variables in a special kind of mathematical model called a stochastic process.

The most famous book on this difficult subject, Stochastic Processes, was published in 1953 by Prof. Joseph Doob, one of the world’s eminent mathematicians. Over a period of years, I corresponded with Doob asking his help or advice on what I thought were highly questionable interpretations of the theory by krigers in what they call the “Theory of Regionalized Variables.” No two krigers ever seem to mean quite the same thing when they say this. Doob was unable to help, saying that the theory needs to be translated into standard mathematics, after which it can be discussed intelligently.

But krigers, most of whom have vested interests in their own proprietary kriging programs or consultancies, fiercely resist any such translation.

They are able to do so because journal editors always send papers about kriging to krigers for peer review.

Indeed, they have twice refused to publish a rigorous mathematical explanation of a special case of Exhibit I, checked and found correct by professional mathematicians. I have in my files violently inimical and even vituperative commentary by some of the world’s best-known krigers rejecting this proof, but demonstrating with painful clarity the incompetence of the reviewers.

Indeed, I stand accused of intellectual terrorism by the Department of Applied Earth Sciences of Stanford University for translating geostatistical gibberish into Doob’s “standard mathematics” and pointing out some serious errors as a result.

The arbitrary sample manipulations and fudge factors so widely used by krigers not well-trained in geology are (or ought to be) frowned upon.

Witness Bre-X Minerals and others of lesser fame. To be sure, Bre-X was not caused by kriging, but it might well have been avoided by applying sounder statistical practices, as Jan Merks evidently did.

Robert Shurtz

San Francisco, Calif.