LETTER TO THE EDITOR — Experience key to reliable geostat data

In response to your editorial “Challenging geostatistics” (T.N.M., Sept. 28-Oct. 4/98), I agree with Jan Merks that the method of ore reserve estimation by kriging within geological limits is incorrect.

Kriging is an oversmoothed estimate, even with closely spaced exploration drilling, and it gets worse as drill hole spacing increases. To compensate, we treat geological boundaries such as rock types as knife-edged grade changes, even though we know their influence to be much more subtle. The idea is to get about the right mix so that the ore tonnage and grade estimate is approximately correct at the cutoffs of interest.

Although widely used, inverse distance weighting isn’t as good. It has often been shown that, with the right variogram, kriging produces a better estimate than inverse distance weighting, though the difference may be small in many cases.

The traditional approach of polygonal areas of influence around each sample is worse still, even though with proper geological bounds, inclusion of internal waste, cutting limits and dilution factors, good agreement with milled ore can be achieved. Kriged estimates have often been shown to give much smaller local errors than polygons. In fact, that was the basis for the development of geostatistics in South Africa in the late 1940s.

The more theoretically correct geostatistical approach to ore reserve estimation is a conditional probability type of estimate (there are many), where the property of additivity of variances is used to predict the mining size blocks within larger blocks, or from oversmoothed kriged estimates. Conditional probability is similar to a weatherman’s prediction of “probability of precipitation.” The problems with conditional probability are that the estimates are poorly understood by operators, that they rely on the correct calculation of a number of geostatistical parameters (all of which vary throughout the deposit) and that the assumptions are still approximations.

My point is that all of the methods we can devise for ore reserve estimation are oversimplifications of a complex problem. Since all ore reserve estimation methods are flawed, it takes a lot of experience to recognize the limitations of each, and this experience factor is the main reason for the failure of any of the estimation methods. My approach is to produce a number of different ore reserve estimates, and to use geostatistics to relate new orebodies to those from my past experience, in order to judge how I would expect each estimate to perform. I prefer that the geologist responsible for the published reserve use kriging within geological outlines because it is widely accepted and simpler to understand. As a check, I rely heavily on conditional probability.

In the past, Jan Merks has criticized kriging as “harmless,” “misleading” or “useless,” but he does not offer a valid alternative. His “mean squared partition” method is an attempt to use sample theory to define error limits (mostly on average grade) within ore outlines. There are two problems with this. He does not provide a method for determining those ore limits from widely spaced exploration drilling, nor does he account for the errors in determining those limits. He states that he prefers geological controls as ore boundaries. So do I, but from my experience, ore limits in most metal deposits must be interpolated from the drill hole data. Without a smoothing method like kriging for interpolating grade limits, ore grades within those limits are systematically overstated.

I have worked on geostatistical ore reserve estimation and grade control in a number of mines over 25 years. On the basis of my geostatistical estimates, six large new mines have been brought into production (Valley Copper, Porgera, Big Bell, Huckleberry, Ernest Henry and George Fisher) and several more have been expanded. I have had many opportunities to check my estimates against production, including mine planning, over a number of years using my own estimates.

On the other hand, when I worked with Jan in the early 1980s, he had little knowledge of ore reserve estimation, geology or geostatistics. He began writing his papers condemning geostatistics only a few years later. Although Jan is a recognized expert on sampling bulk solids, his lack of experience in ore reserve estimation shows in his papers. For example, in his 1991 paper on the mean squared partition method, he gives an example of determining error limits on the average grade of a mining drift from bulk samples. This is not an ore reserve estimation.

Gary Raymond, P.Eng.

Summerland, B.C.