Sampling Patterns
The idea of extension variance and combining elementary extension variances allows us to compare the efficiencies of sampling patterns (orsampling geometries ). The are three broad types of sampling geometry as illustrated in figure 8.3.
Figure 8.3 Sampling patterns and clustering (‘sampling geometries’)
Considering each of these separately: Random Pattern
Random Pattern
Random sampling would be unusual in mining practice, however there are instances where old data, or intersections in a vein at depth (due to lack of control on deep holes) might follow this pattern.
In order to estimate the average value Z V ( ) overV we take the average of N
samples Z x( )i randomly scattered withinV . We will assume that our samples
Z x( )i have point support (i.e. that they are very small in relation to the domain being estimated). It can be shown that the variance of the error is:
σ E
N D o V
2
=
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where D o V 2( | )
is simply the dispersion variance of points in the domain we are estimating, i.e. the variance of the samples.
Random Stratified Grid (RSG) Random Stratified Grid (RSG)
This time,V is divided into N similar sub zones. Within each sub zone, a sample is taken at a random location. Hence this pattern is ‘less random’ than a truly random sampling. Intersections in a vein at depth that are intended to be on a regular grid in the plane of the vein might follow this pattern simply as a consequence of hole deviations. RSG's are also sometimes used in geochemical soil surveys.
In this case it can be shown that the variance of estimation has the same form as for the purely random case, and is:
σ E N D o v 2
=
1 2 ( | ) Note that D o V 2 ( | ) is replaced here by D o v2( | ). Since we know from the Krige's relationship that D o V 2
( | )is always greater than D o v2
( | ):
D o V 2( | )
−
D o v 2( | )=
D v V 2( | )≥
0So, the estimation variance for an RSG will always be lower than for a purely random grid, since:
σ E
N D o v N D o V
2
=
1 2( | )<
1 2( | )This shows that the Random Stratified Grid is always more efficient than a purelyrandom pattern. Regular Grid
Regular Grid
We have seen that in this case the estimation variance can be approximated by the principle ofcombination of elementary extension variances , i.e. it can be approximated by dividing the extension variance of a central sample in a unit block by N (the number of informed elementary blocks):
σ E σ e
N
2
=
2We can calculate (using auxiliary functions) the estimation variance of a given grid, for a specified domainV and known variogram model. Delfiner (1979) gives an comparison for a linear variogram model, showing that for a given situation, the ratio between the estimation variance for an RSG and a regular grid is about 2.14. In other words, a regular grid is better than twice as efficient as an RSG (for the same number of samples).
We know that the RSG is always more efficient than a totally random pattern. In fact, Delfiner gives a comparison for the three cases when the domainVis a square.
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He expresses the estimation variances in terms of the sample variance
(σ 2
=
D o V 2[ | ])and the number of samples N :
Table 6.2 Estimation variances for the average
Table 6.2 Estimation variances for the average grade overgrade over a square (with a linear variogram
a square (with a linear variogram andand N N samples)) samples))
Random
Random Pattern Pattern RSG RSG Regular Regular PatternPattern
σ 2 N σ 2 3 2 N / σ 2 3 2 2 14.
×
N / We can appreciate the benefit of a regular sampling pattern from this table. The variance is reduced considerably as our sampling geometry becomes more regular. A regular sampling pattern exploits the spatial correlation of the grades: putting twosamples very close together is producing redundant information, and (given a limited number of samples) implies that some other part of the domain must be under-sampled.
Similar results can be determined for a bounded variogram model (say, spherical), but the estimation variance now depends upon the rangeaof the variogram. It can be demonstrated that the regular grid will still perform better than the RSG, but the advantage becomes reduced as the size of the grid square becomes large relative to the range of the variogram.
Again, this accords with common sense: as the unit cell in our grid becomes much larger than the range of the variogram, the influence of a sample becomes, relatively, quite localised. Consequently, the strategic superiority of the central location declines. In fact, asabecomes small relative to the dimensions of our grid cell, the situation more closely approaches that of a pure nugget effect model (i.e.
a
→
0 ).Delfiner gives the following example, in which he tabulated, for various cell sizes, the dispersion variance of points in a cellv(the estimation variance of an RSG is proportional to this) and the extension variance of the same unit cell (using a spherical model) as seen in the following table.
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Table 6.3 Table 6.3 Dimension of Dimension of Grid in Units Grid in Units of Range ( of Range ( a a)) D o v2( | ) σ e o v 2( , ) ratioratio 0.15a 0.118 0.056 2.107 0.2a 0.115 0.074 2.094 0.4a 0.31 0.15 2.067 0.6a 0.45 0.235 1.915 0.8a 0.56 0.32 1.75 1.0a 0.66 0.41 1.61 1.5a 0.81 0.65 1.25 2.0a 0.88 0.80 1.1 3.0a 0.94 0.92 1.02 5.0a 0.977 0.96 1.017 8.0a 0.9906 0.99 ≈1.00
Note that, after the length of the side of a cell exceeds the range of the variogram, the advantage of the regular grid falls away rapidly. Another way of looking at this is that the estimation variance associated with a single cell climbs very quickly if the grid spacing is too wide.
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