The parent block size is one of the most important parameters which impacts the quality of grade estimates in a resource model. A block size that is too small results in the grade estimates being oversmoothed, giving an incorrect grade-tonnage relationship. Ideally, we’d like to produce a locally accurate estimate at the smallest block size we can to give adequate resolution of the grades in the model – but how do we decide what block size to use? And once we’ve decided on a block size, what about other estimation parameters such as how many samples we use to estimate the grade of a block?
This is where kriging neighbourhood analysis (KNA) comes in. A KNA provides a quantitative method of testing different estimation parameters (e.g. block size) and, by assessing their impact on the quality of the resultant estimate, select the optimal value for each parameter. This will be dependent on several factors within the deposit; the inherent variability, the ranges of grade continuity, anisotropy and the data spacing. The variogram mathematically represents these factors and is a critical input for a KNA.
Supervisor’s existing KNA feature allows the user to run a KNA, but only on a single block, which is often placed in the well supported regions of the domain. This has been enhanced in the latest version of Supervisor to allow a KNA to be run on multiple blocks within a domain, providing more confidence in the parameters being selected. Supervisor analyses the drillhole data and generates block centroids for a domain based on the minimum and maximum coordinates of the data. The user specifies tolerance distances from the data, in the X, Y and Z directions, to restrict which blocks are included in the KNA results to avoid excessive extrapolation away from the data (which always results in poor kriging statistics). The blocks generated for each block size can be viewed in the 3D viewer (Figure 1).
Statistics for KNA are calculated by Supervisor using ordinary kriging. This is a linear method of estimation and therefore only the position of samples relative to each other and the position of the samples relative to the block (or point) being estimated are required to generate the KNA statistics.
The statistics generated for KNA measure conditional bias. This refers to the ‘degree of oversmoothing’ (i.e. reduction in the variance of grades) in the block estimates compared to the theoretical true variance of grades at that block size. The KNA aims to determine the parameters (block size, number of samples, search radius and discretisation) that minimise the conditional bias in the estimate, along with ensuring that not too many negative weights are generated.
There are two conditional bias statistics used for optimisation:
- Kriging efficiency (KE), which measures the effectiveness of the kriged estimate to reproduce the local block grade accurately.
- Slope of regression or conditional bias slope (SLOPE), which summarises the degree of oversmoothing of high and low grades.
The following variance statistics, which are outputs from the kriging equations, are used to establish the kriging efficiency and slope of regression for each block:
- Kriging variance (KV) – is calculated from the variogram and the kriging weights assigned during estimation. It provides a relative measure of accuracy of the local kriged estimate with respect to data coverage.
- LaGrange multiplier (µ) – increases with extrapolation, poor data coverage and/or clustering. This is calculated when the kriging weights are determined.
- Block variance (BV) – measures the degree of difference (variance) between block grades. BV is dependent on the block size and range of continuity, and will increase as block size decreases due to the volume-variance effect.
The kriging efficiency measures the effectiveness of the kriged estimate to reproduce the local block grade accurately. It is calculated by comparing the kriging variance of the block with the theoretical variance of the block.
When the kriging variance is small relative to the block variance then the kriging efficiency approaches a value of one. When the kriging variance is high and dominates the block variance (as would be the case for poorly estimated blocks) then the kriging efficiency will be low (sometimes even negative).
Low kriging efficiency indicates a high degree of oversmoothing. Conversely, high kriging efficiency indicates a low degree of oversmoothing (Figure 2).
Kriging efficiency is often reported as a percentage with the optimal value being 100% (or 1).
The slope of regression summarises the degree of oversmoothing of high and low grades. This slope is equivalent to the regression slope of the estimated block grades against the corresponding true, but unknown, grades (Figure 3). The slope statistic is calculated as:
The LaGrange multiplier will be small when there is good data coverage, no extrapolation and limited clustering. In this situation the LaGrange multiplier is negligible compared with the rest of the equation and the slope statistic approaches one.
However, when the LaGrange multiplier is large (high degree of extrapolation, poor data coverage and/or clustering), then the LaGrange multiplier dominates the equation and the slope statistic tends towards a half.
A slope close to one indicates that the regression between the estimated and true grades is likely to be very good, meaning there is limited oversmoothing. In this case it is likely that the grade tonnage relationship above cut-off is realistic.
Conversely, low slope values indicate that there is oversmoothing and hence a poor relationship between the estimated and actual block grades. In this instance it is unlikely that you will be able to accurately report selective estimates above a cut-off.
The conditional bias statistics can be generated for any combination of variogram and estimation parameters to test various parameters and determine the optimal estimation parameters. Once the statistics are determined for each value of a parameter, scenarios can be compared. Ideally the optimal result is a slope of one and a kriging efficiency of 100%; however, this is never achievable in practice. More typical results are slopes of greater than 0.9 and kriging efficiencies in the order of 80% to 90% for well-informed areas.
As multiple blocks are being tested by Supervisor, rather than providing a single result, the results are graphed as a box and whisker plot to show not only the average kriging efficiency and slope of regression, but also the range of results (Figure 4).
In areas of sparse drilling such as during the earlier exploration stages, or in domains with short ranges of grade continuity or elevated nugget effect (e.g. gold), the results will be lower. Additionally, in narrow domains the results may be poor due to the lack of data in the third dimension. This method can still be used in a relative sense however, to determine which scenario provides the better results.
Negative weights can be generated during ordinary kriging due to screening of samples behind one another. Whilst mathematically correct, too many negative weights are not ideal and can cause anomalies. Parameters should be selected to minimise the number of negative weights. The sum of the negative weights and percentage of negative weights for each block is calculated by Supervisor and presented as a box and whisker plot (Figure 5), similar to the kriging efficiency and slope of regression.