It has been bought to our attention that under certain conditions, deemed highly unlikely, that the Lagrange multiplier may return valid negative values, which makes the use of the ‘absolute’ condition in the equation for the slope of regression incorrect.
The conditions triggering this are domains which have extremely low nugget values combined with long ranges and close-spaced drilling. In these circumstances the Lagrange values will be negligible in any case, and therefore will have little impact on the final results obtained when using drill hole spacing analysis.
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The updated version of Supervisor (184.108.40.206) can be downloaded from the Supervisor page
How much drilling do you need to be confident that predicted grades and production targets will obtainable? Too much drilling and you are wasting valuable time, money and resources that could be used elsewhere. Too little drilling and the prediction of grades and production targets becomes extremely difficult because of the lack of data.
The purpose of drill hole spacing analysis (DHSA) is to determine the optimal drill hole spacing for a particular deposit. This will be dependent on several factors within the deposit; the inherent variability, the ranges of grade continuity and anisotropy. The variogram mathematically represents these factors and is a critical input for DHSA.
With the release of Version 8.6, Supervisor now allows users to quickly and easily conduct DHSA on their deposits.
To calculate statistics for DHSA Supervisor uses ordinary kriging. This is a linear method of estimation and therefore grades are not used to generate the statistics for drill hole spacing. In linear estimation techniques 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.
The statistics generated for DHSA measure conditional bias. This refers to the ‘degree of over smoothing’ (reduction in variance of grades) in the estimate compared to the true variance of grade in the deposit. The DHSA aims to determine the drilling grid which minimises the conditional bias in the estimate.
There are two conditional bias statistics used for optimisation:
- Kriging efficiency (KE), which measures the effectiveness of the kriging estimate to reproduce the local block grade accurately.
- Slope of regression or conditional bias slope (SLOPE), which summarises the degree of over smoothing of high and low grades.
- The following variance statistics are used to establish the kriging efficiency and slope of regression:
- 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. It is dependant on block size and range and will increase as block size decreases due to the volume variance effect.
The kriging efficiency measures the effectiveness of the kriging 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 over smoothing. Conversely, high kriging efficiency indicates a low degree of over smoothing (Figure 1).
Kriging efficiency is often reported as a percentage with the optimal value being 100% (1).
The slope of regression summarises the degree of over smoothing 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 2). 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 actual grades is likely to be very good, meaning there is limited over smoothing. In this case it is likely that the grade tonnage relationship above cut-off is realistic.
Conversely, low slope values indicate that there is over smoothing 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 drill grid, variogram and estimation parameters to test various drill spacings.
While the total estimate can be run each time and the results averaged, this is often impractical. An alternative is to create a single block, or a small group of blocks, centred in the area you want to test. This will give you a representative result for this area. It is recommended that several locations are tested.
Once the statistics are determined for each drill hole grid scenarios can be compared (Figure 3). 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%.
In areas of sparse drilling such as during the exploration stage, or in domains with short range grade continuity (e.g. gold), the results will be lower. Additionally, in narrow domains the results will be poor due to the lack of data in the third dimension. This method can still be used in a relative sense to determine which scenario provides the better results.
There is an example of how to complete DHSA located here.