Silicon Dale

The true nugget effect ?

One of the most important assumptions behind geostatistics is that the semivariogram uniquely describes the spatial covariance properties of a data set (a correlogram can be used instead, of course, but the mathematical relationship between the two is close enough). Implicit in its computation is that the only factor controlling the semivariogram value is the lag distance between any two points in the region under investigation. Geostatistical practitioners normally assume that any inhomogeneity can be removed by defining homogeneous zones, and for the purposes of this discussion we shall accept that this can be done (though it's a big assumption and often not true). The very real problems of non-stationarity have been discussed in another Silicondale column and therefore shall also be ignored for the purposes of this discussion.

Nevertheless, in deposits of many minerals (especially gold, PGMs, and diamonds) it is found that there will be many lower-grade assays and relatively few high-grade assays. This can also cause problems in applying geostatistical methods, due to apparent highly skewed data distributions. For this reason, a number of methods have been developed including lognormal kriging, indicator kriging, and multigaussian kriging, in attempts to make the skewness more tractable.

However, the real problem is perhaps a little more serious than these methods allow. Geostatisticians have long recognised that there is often a 'proportional effect' in which higher grades are associated with higher variances and hence with a higher sill on the variogram. It has been claimed that the use of relative variograms can help to overcome this proportional effect, in which higher grades are associated with higher variance. Relative variograms, with variance adjusted for grade, may allow the sill variances for high and low grades to be harmonised. But there is another effect which cannot be handled (or allowed for) by geostatistical methods and which (perhaps as a direct result) has remained unmentioned. This is that not only the variance but also the range can be dependent on grade. It is evident that a low-grade sample is likely to be representative of a much larger surrounding volume than a high-grade sample (or in the extreme case a nugget, where the range is reduced to the radius of the nugget itself). There is no way that computation of relative variograms can overcome an inverse relationship between grade and range.

That this can be a very real practical issue is illustrated by a known problem with multiple indicator kriging. Here, a series of cutoff grades is defined, and a set of 0/1 indicators defined for each cutoff. It is then possible to compute variograms and fit variogram models for each cutoff. There is nothing constraining these variogram models to have the same parameters for each and every cutoff (indeed the models used may themselves vary - for example, exponential for one cutoff, spherical for another, and with different amounts and directions of anisotropy. In particular it should be noted that the range may vary. Unfortunately, in the formulation of the multiple indicator kriging method, such differences in variograms for each cutoff can lead to logical inconsistencies (‘order-relation problems'). Indeed, this is but one of a number of reasons why indicator kriging itself is in my opinion seriously flawed (but that's another story - see ESCA vol.17, no.1, Sept 2001).

I would propose that any solution to this problem caused by the true nugget effect must allow the actual grade of samples to influence the estimation method. In geostatistics the only control is provided by the spatial distribution of data points and a variogram model (or set of variogram models) which are completely unaffected by local grade variations. In conventional (i.e. non- spatial) statistics, analogous problems have been addressed in recent years by the development of adaptive estimators. Such estimators can adapt to the local properties of a data set, to give much more reliable estimates than are offered by the rigid traditional methods. How such adaptive estimation techniques might best be translated into spatial estimation is not yet clear, though their relationship with nonparametric statistics suggests that a nonparametric approach would be more appropriate than the 'classical' least-squares based geostatistical methods.

One or two of these methods were explored in my book Nonparametric Geostatistics. This is now out of print, but I have 30 remaining copies available. A copy of Fortran source code which includes examples of such estimation methods is available for free download as part of the nonparametric estimation package on www.silicondale.com.

Stephen Henley
Matlock, England
steve@silicondale.co.uk
www.SiliconDale.com

Copyright © 2001 Stephen Henley
The true nugget effect ?: Earth Science Computer Applications, v.17,no.5,p.1-2