The Problem of Resource Estimation
Sampling data from drill holes, channels etc. provide the geologist and mining engineer with fragmentary information. The problem we face when performing resource estimation is to obtain an idea of the grade of the whole deposit, or of specific blocks of ground within the deposit. The only solution to this problem is to makeestimates .
No data processing, mathematics or computer program can ever tell us what the grade of an un-sampled point in our deposit is. There is always uncertainty:
1. Between sampled locations (refected by the spatial component of the variogram, and
2. Atthe samples themselves (this is reflected in the nugget effect, as we have discussed earlier).
The variogram is thus a model for spatial uncertainty, and a vital input to designing sensible estimation approaches.
So we must make estimates, and the aim is surely to make the ‘best’ estimates we can, to use our data efficiently, i.e. to get the most out of our information (the samples). We will look at the issue of ‘best’ shortly; first, lets consider the idea of estimation by interpolation in a bit more detail.
Most geologists are comfortable with the idea of using some sort of weighted average of sample values to estimate blocks in a mineral deposit. A ‘classical’ method for doing this is the use of inverse distance weighting (IDW).
Chapter
Chapter
9
9
Uncertainty Uncertainty No data processing, mathematics or computer program can ever tell us what the grade of an un-sampled point in our deposit is. There is always uncertainty between sampled locations.
CH 10 –
CH 10 – NON-LINEAR ESTIMATIONNON-LINEAR ESTIMATION
Quantitative Group Short Course Manual
In IDW methods, the weighting coefficients (or more simply the ‘weights’) applied to a sample are a function of the position of the sample relative to the block being estimated. Samples close to the block get higher weights than those far away. Some general idea of the variability of the mineralisation may be introduced (by using higher powers, for instance, weighting by the inverse of the distance squared, or cubed etc.), but no real reference is made to the variability of the particular orebody under consideration.
That IDW should be more appealing to mining geologists than polygonal methods is not surprising—grades in mineral deposits are, almost invariably, correlated in space, often up to quite large distances. Because of this correlation, samples external to the block being estimated can reveal information about the block grade, and thus improve estimation accuracy compared to polygonal methods. Methods that use weighted averages of local data thus represent a significant improvement over polygonal methods.
However, there is no reason that IDW weighting should be applicable to ore deposits, or, putting it another way—to quote A.G. Royle—“while the inverse of the distance squared may have some physical relation to the speed of light in a vacuum, it has nothing whatsoever to do with the distribution of grades in a
mineral deposit”. It has never been made clear by any of its proponents why any particular power of the inverse distance should be used in resource estimation practice.
What Do We Want From An Estimator? What Do We Want From An Estimator?
Whatdo we desire of an estimator? At the very least we require anaccurate estimate, i.e. we wish our estimates to be (on average) as close to the ‘true’ grades as possible. The accuracy of estimates depend on a number of factors, i.e. those that affect the
estimation varianceσ e
2
introduced in the previous chapter:
1. The number of samples and the quality of the data for each sample. The quality may vary quite significantly from one sample to another. Our
estimator should not necessarily grant the same importance to each sample used for the estimate.
2. The geometry of samples in the deposit. In particular, clustering of samples may make some sampling information ‘redundant’, at least in part. In general, an even distribution of samples in the deposit achieves better coverage and gives more information than an equivalent number of samples that are locally clustered.
3. The distance between a sample and the area we wish to estimate. If we wish to estimate a particular block, it is natural to place more weight on
samples close to the block of interest than on more distant samples. Similarly, if we wish to make point estimates (for example, of a thickness variable) we expect our estimator to be exact at the point where we have
data, more reliable close to sample points, and to deteriorate as distance to the nearest sample increases.
Interpolation Interpolation
Methods that use weighted averages of local data thus
represent a significant improvement over polygonal methods.
CH 10 –
CH 10 – NON-LINEAR ESTIMATIONNON-LINEAR ESTIMATION
Quantitative Group Short Course Manual
4. The spatial continuity of the interpolated variables. We should require more than an arbitrary falling away of weight as distance to the block or point being estimated increases, such as provided by IDW. We should also require that the spatial variability of the variables is incorporated into our estimates. Variables with very smooth variations (for example the top of a gently deformed geological horizon) should not be weighted in the same manner as variables with more erratic spatial fluctuation, like metal grades. We desire that our estimator isunbiased , i.e. the average estimation error should be
zero (so that the average of the estimates should always equal the average of the true grades).
We would also like to have an index of thereliability of the estimates.